-
1
Chapter 2. New Study of Magic Squares of Order 4: Kanji Setsuda
Section 2. New Algebraic Study of Pan-Magic Squares 4x4 #0.
Classical Concept of "Pan-Magic Squares" of Order 4 Let's study
about the Pan-diagonal type of Magic Squares of Order 4 here. I
know there are various kinds of definition of this type, but I like
to start with such a simple explanation about our classical concept
for the 'Pan-magic Shift' as follows. I want you to see the next
diagrams that demonstrate how the Basic Form can be transformed
into other forms of order 4. We want those two types of
transformation to be always possible for making the 'Magic Squares
of Order 4' with the constant sums of their rows, columns and
primary diagonals every time. How should we define the Basic
Conditions for that type at the first stage? Fig. 1 Fig. 2 Diagrams
for the 'Pan-magic Shifts' .--.--.--.--. .--.--.--.--. of Pan-magic
Squares of Order 4 | 1| 2| 3| 4| 1 1| 2| 3| 4| 1| |--+--+--+--|
|--+--+--+--| Type 1: | 5| 6| 7| 8| 5 5| 6| 7| 8| 5| (1) Select the
column at the left side. |--+--+--+--| |--+--+--+--| | 9|10|11|12|
9 9|10|11|12| 9| (2) Move it with the 4 numbers beyond
|--+--+--+--| |--+--+--+--| the wall to the right end.
|13|14|15|16|13 13|14|15|16|13| '--'--'--'--' '--'--'--'--' (3) Put
it there for a new column. Fig. 1 Fig. 5 .--.--.--.--. 1 2 3 4 | 1|
2| 3| 4| .--.--.--.--. |--+--+--+--| | 5| 6| 7| 8| Type 2: | 5| 6|
7| 8| |--+--+--+--| (1) Select the row on the top floor.
|--+--+--+--| | 9|10|11|12| | 9|10|11|12| |--+--+--+--| (2) Move it
down with the 4 numbers |--+--+--+--| |13|14|15|16| to the bottom
end under the ground. |13|14|15|16| |--+--+--+--| '--'--'--'--' |
1| 2| 3| 4| (3) Put it there for a new row. 1 2 3 4 '--'--'--'--'
How many different forms can we make by those two types of
Pan-magic Shift, if we can repeat any type and combine them in any
order? How many possible primary diagonals should we define to add
up to the constant? ** Diagrams of the Pan-magic Shifts for
Pan-magic Squares of Order 4 ** ** Made by Kanji Setsuda on July 7,
2010 with MacOSX & Xcode 3.1.2 **
------------------------------------------------------------------------------
Fig. 1 For Rows and Columns: | Fig. 2 For Rows and Columns:
.--.--.--.--. n1+n2+n3+n4=C; | .--.--.--.--. n2+n3+n4+n1=C; | 1| 2|
3| 4| 1 n5+n6+n7+n8=C; | | 2| 3| 4| 1| 2 n6+n7+n8+n5=C;
|--+--+--+--| n9+n10+n11+n12=C; | |--+--+--+--| n10+n11+n12+n9=C; |
5| 6| 7| 8| 5 n13+n14+n15+n16=C; | | 6| 7| 8| 5| 6
n14+n15+n16+n13=C; |--+--+--+--| n1+n5+n9+n13=C; | |--+--+--+--|
n2+n6+n10+n14=C; | 9|10|11|12| 9 n2+n6+n10+n14=C; | |10|11|12| 9|10
n3+n7+n11+n15=C; |--+--+--+--| n3+n7+n11+n15=C; | |--+--+--+--|
n4+n8+n12+n16=C; |13|14|15|16|13 n4+n8+n12+n16=C; | |14|15|16|13|14
n1+n5+n9+n13=C; '--'--'--'--' Primary diagonals: | '--'--'--'--'
Primary diagonals: 1 2 3 4 n1+n6+n11+n16=C; | 2 3 4 1
n2+n7+n12+n13=C; n4+n7+n10+n13=C; | n1+n8+n11+n14=C;
-
2
------------------------------------------------------------------------------
Fig. 3 For Rows and Columns: | Fig. 4 For Rows and Columns:
.--.--.--.--. n3+n4+n1+n2=C; | .--.--.--.--. n4+n1+n2+n3=C; | 3| 4|
1| 2| 3 n7+n8+n5+n6=C; | | 4| 1| 2| 3| 4 n8+n5+n6+n7=C;
|--+--+--+--| n11+n12+n9+n10=C; | |--+--+--+--| n12+n9+n10+n11=C; |
7| 8| 5| 6| 7 n15+n16+n13+n14=C; | | 8| 5| 6| 7| 8
n16+n13+n14+n15=C; |--+--+--+--| n3+n7+n11+n15=C; | |--+--+--+--|
n4+n8+n12+n16=C; |11|12| 9|10|11 n4+n8+n12+n16=C; | |12| 9|10|11|12
n1+n5+n9+n13=C; |--+--+--+--| n1+n5+n9+n13=C; | |--+--+--+--|
n2+n6+n10+n14=C; |15|16|13|14|15 n2+n6+n10+n14=C; | |16|13|14|15|16
n3+n7+n11+n15=C; '--'--'--'--' Primary diagonals: | '--'--'--'--'
Primary diagonals: 3 4 1 2 n3+n8+n9+n14=C; | 4 1 2 3
n4+n5+n10+n15=C; n2+n5+n12+n15=C; | n3+n6+n9+n16=C;
------------------------------------------------------------------------------
Fig. 5 For Rows and Columns: | Fig. 6 For Rows and Columns:
.--.--.--.--. n5+n6+n7+n8=C; | .--.--.--.--. n6+n7+n8+n5=C; | 5| 6|
7| 8| 5 n9+n10+n11+n12=C; | | 6| 7| 8| 5| 6 n10+n11+n12+n9=C;
|--+--+--+--| n13+n14+n15+n16=C; | |--+--+--+--| n14+n15+n16+n13=C;
| 9|10|11|12| 9 n1+n2+n3+n4=C; | |10|11|12| 9|10 n2+n3+n4+n1=C;
|--+--+--+--| n5+n9+n13+n1=C; | |--+--+--+--| n6+n10+n14+n2=C;
|13|14|15|16|13 n6+n10+n14+n2=C; | |14|15|16|13|14 n7+n11+n15+n3=C;
|--+--+--+--| n7+n11+n15+n3=C; | |--+--+--+--| n8+n12+n16+n4=C; |
1| 2| 3| 4| 1 n8+n12+n16+n4=C; | | 2| 3| 4| 1| 2 n5+n9+n13+n1=C;
'--'--'--'--' Primary diagonals: | '--'--'--'--' Primary diagonals:
5 6 7 8 n5+n10+n15+n4=C; | 6 7 8 5 n6+n11+n16+n1=C;
n8+n11+n14+n1=C; | n5+n12+n15+n2=C;
------------------------------------------------------------------------------
Fig. 7 For Rows and Columns: | Fig. 8 For Rows and Columns:
.--.--.--.--. n7+n8+n5+n6=C; | .--.--.--.--. n8+n5+n6+n7=C; | 7| 8|
5| 6| 7 n11+n12+n9+n10=C; | | 8| 5| 6| 7| 8 n12+n9+n10+n11=C;
|--+--+--+--| n15+n16+n13+n14=C; | |--+--+--+--| n16+n13+n14+n15=C;
|11|12| 9|10|11 n3+n4+n1+n2=C; | |12| 9|10|11|12 n4+n1+n2+n3=C;
|--+--+--+--| n7+n11+n15+n3=C; | |--+--+--+--| n8+n12+n16+n4=C;
|15|16|13|14|15 n8+n12+n16+n4=C; | |16|13|14|15|16 n5+n9+n13+n1=C;
|--+--+--+--| n5+n9+n13+n1=C; | |--+--+--+--| n6+n10+n14+n2=C; | 3|
4| 1| 2| 3 n6+n10+n14+n2=C; | | 4| 1| 2| 3| 4 n7+n11+n15+n3=C;
'--'--'--'--' Primary diagonals: | '--'--'--'--' Primary diagonals:
7 8 5 6 n7+n12+n13+n2=C; | 8 5 6 7 n8+n9+n14+n3=C; n6+n9+n16+n3=C;
| n7+n10+n13+n4=C;
------------------------------------------------------------------------------
Fig. 9 For Rows and Columns: | Fig.10 For Rows and Columns:
.--.--.--.--. n9+n10+n11+n12=C; | .--.--.--.--. n10+n11+n12+n9=C; |
9|10|11|12| 9 n13+n14+n15+n16=C; | |10|11|12| 9|10
n14+n15+n16+n13=C; |--+--+--+--| n1+n2+n3+n4=C; | |--+--+--+--|
n2+n3+n4+n1=C; |13|14|15|16|13 n5+n6+n7+n8=C; | |14|15|16|13|14
n6+n7+n8+n5=C; |--+--+--+--| n9+n13+n1+n5=C; | |--+--+--+--|
n10+n14+n2+n6=C; | 1| 2| 3| 4| 1 n10+n14+n2+n6=C; | | 2| 3| 4| 1| 2
n11+n15+n3+n7=C; |--+--+--+--| n11+n15+n3+n7=C; | |--+--+--+--|
n12+n16+n4+n8=C; | 5| 6| 7| 8| 5 n12+n16+n4+n8=C; | | 6| 7| 8| 5| 6
n9+n13+n1+n5=C; '--'--'--'--' Primary diagonals: | '--'--'--'--'
Primary diagonals: 9 10 11 12 n9+n14+n3+n8=C; | 10 11 12 9
n10+n15+n4+n5=C; n12+n15+n2+n5=C; | n9+n16+n3+n6=C;
------------------------------------------------------------------------------
Fig.11 For Rows and Columns: | Fig.12 For Rows and Columns:
.--.--.--.--. n11+n12+n9+n10=C; | .--.--.--.--. n12+n9+n10+n11=C;
|11|12| 9|10|11 n15+n16+n13+n14=C; | |12| 9|10|11|12
n16+n13+n14+n15=C; |--+--+--+--| n3+n4+n1+n2=C; | |--+--+--+--|
n4+n1+n2+n3=C; |15|16|13|14|15 n7+n8+n5+n6=C; | |16|13|14|15|16
n8+n5+n6+n7=C; |--+--+--+--| n11+n15+n3+n7=C; | |--+--+--+--|
n12+n16+n4+n8=C; | 3| 4| 1| 2| 3 n12+n16+n4+n8=C; | | 4| 1| 2| 3| 4
n9+n13+n1+n5=C;
-
3
|--+--+--+--| n9+n13+n1+n5=C; | |--+--+--+--| n10+n14+n2+n6=C; |
7| 8| 5| 6| 7 n10+n14+n2+n6=C; | | 8| 5| 6| 7| 8 n11+n15+n3+n7=C;
'--'--'--'--' Primary diagonals: | '--'--'--'--' Primary diagonals:
11 12 9 10 n11+n16+n1+n6=C; | 12 9 10 11 n12+n13+n2+n7=C;
n10+n13+n4+n7=C; | n11+n14+n1+n8=C;
------------------------------------------------------------------------------
Fig.13 For Rows and Columns: | Fig.14 For Rows and Columns:
.--.--.--.--. n13+n14+n15+n16=C; | .--.--.--.--. n14+n15+n16+n13=C;
|13|14|15|16|13 n1+n2+n3+n4=C; | |14|15|16|13|14 n2+n3+n4+n1=C;
|--+--+--+--| n5+n6+n7+n8=C; | |--+--+--+--| n6+n7+n8+n5=C; | 1| 2|
3| 4| 1 n9+n10+n11+n12=C; | | 2| 3| 4| 1| 2 n10+n11+n12+n9=C;
|--+--+--+--| n13+n1+n5+n9=C; | |--+--+--+--| n14+n2+n6+n10=C; | 5|
6| 7| 8| 5 n14+n2+n6+n10=C; | | 6| 7| 8| 5| 6 n15+n3+n7+n11=C;
|--+--+--+--| n15+n3+n7+n11=C; | |--+--+--+--| n16+n4+n8+n12=C; |
9|10|11|12| 9 n16+n4+n8+n12=C; | |10|11|12| 9|10 n13+n1+n5+n9=C;
'--'--'--'--' Primary diagonals: | '--'--'--'--' Primary diagonals:
13 14 15 16 n13+n2+n7+n12=C; | 14 15 16 13 n14+n3+n8+n9=C;
n16+n3+n6+n9=C; | n13+n4+n7+n10=C;
------------------------------------------------------------------------------
Fig.15 For Rows and Columns: | Fig.16 For Rows and Columns:
.--.--.--.--. n15+n16+n13+n14=C; | .--.--.--.--. n16+n13+n14+n15=C;
|15|16|13|14|15 n3+n4+n1+n2=C; | |16|13|14|15|16 n4+n1+n2+n3=C;
|--+--+--+--| n7+n8+n5+n6=C; | |--+--+--+--| n8+n5+n6+n7=C; | 3| 4|
1| 2| 3 n11+n12+n9+n10=C; | | 4| 1| 2| 3| 4 n12+n9+n10+n11=C;
|--+--+--+--| n15+n3+n7+n11=C; | |--+--+--+--| n16+n4+n8+n12=C; |
7| 8| 5| 6| 7 n16+n4+n8+n12=C; | | 8| 5| 6| 7| 8 n13+n1+n5+n9=C;
|--+--+--+--| n13+n1+n5+n9=C; | |--+--+--+--| n14+n2+n6+n10=C;
|11|12| 9|10|11 n14+n2+n6+n10=C; | |12| 9|10|11|12 n15+n3+n7+n11=C;
'--'--'--'--' Primary diagonals: | '--'--'--'--' Primary diagonals:
15 16 13 14 n15+n4+n5+n10=C; | 16 13 14 15 n16+n1+n6+n11=C;
n14+n1+n8+n11=C; | n15+n2+n5+n12=C;
------------------------------------------------------------------------------
We can make 16 different forms in all as shown above from the Fig.1
including itself. No more or no less. Can you see every set of the
conditions for rows and columns always have the same content in
common? Yes. The contents are kept unchanged, while the equations
and variables themselves appear in different orders. How about the
conditions of our new primary diagonals I collected here as
follows? ** Collective List of all the Possible Primary diagonals:
C=34 ** n1+n6+n11+n16=C ...pd11; | n4+n7+n10+n13=C ...pd12;
n2+n7+n12+n13=C ...pd21; | n1+n8+n11+n14=C ...pd22; n3+n8+n9+n14=C
...pd31; | n2+n5+n12+n15=C ...pd32; n4+n5+n10+n15=C ...pd41; |
n3+n6+n9+n16=C ...pd42; n4+n5+n10+n15=C ...pd51; | n1+n8+n11+n14=C
...pd52; n1+n6+n11+n16=C ...pd61; | n2+n5+n12+n15=C ...pd62;
n2+n7+n12+n13=C ...pd71; | n3+n6+n9+n16=C ...pd72; n3+n8+n9+n14=C
...pd81; | n4+n7+n10+n13=C ...pd82; n3+n8+n9+n14=C ...pd91; |
n2+n5+n12+n15=C ...pd92; n4+n5+n10+n15=C ...pd101; | n3+n6+n9+n16=C
...pd102; n1+n6+n11+n16=C ...pd111; | n4+n7+n10+n13=C ...pd112;
n2+n7+n12+n13=C ...pd121; | n1+n8+n11+n14=C ...pd122;
n2+n7+n12+n13=C ...pd131; | n3+n6+n9+n16=C ...pd132; n3+n8+n9+n14=C
...pd141; | n4+n7+n10+n13=C ...pd142; n4+n5+n10+n15=C ...pd151; |
n1+n8+n11+n14=C ...pd152; n1+n6+n11+n16=C ...pd161; |
n2+n5+n12+n15=C ...pd162; The new conditions have come all at the
first four cases and the rest of conditions appeared with the same
contents as the first eight. You can certainly find repetition.
-
4
We really need those eight conditions for diagonals pd11~pd42.
When we have all of them from the beginning, we can always compose
any new magic squares of order 4 by our Pan-magic Shifts anytime.
It is these eight that surely make 16 squares. Let's call the
"Pan-diagonals" for the eight above. Let's add all the eight
conditions above to our definition list at the first stage. #1. New
Concept of Pan-Diagonal Magic Square of Order 4 Let's start now
simply with these diagrams and simultaneous equations as follows.
[Basic Form and Basic Definitions] .--.--.--.--. * Rows and Columns
* |n1|n2|n3|n4| n1+ n2+ n3+ n4 = C ... rw1 |--+--+--+--| n5+ n6+
n7+ n8 = C ... rw2 |n5|n6|n7|n8| n9+n10+n11+n12 = C ... rw3
|--+--+--+--| n13+n14+n15+n16 = C ... rw4 |n9|10|11|12| n1+ n5+
n9+n13 = C ... cl1 |--+--+--+--| n2+ n6+n10+n14 = C ... cl2
|13|14|15|16| n3+ n7+n11+n15 = C ... cl3 '--'--'--'--' n4+
n8+n12+n16 = C ... cl4 [Extended Space, Moving Frame and
Pan-diagonals] .--.--------.--. 11 12|n9 10 11 12|n9|10 11
.--.--.--------.--.--. 15|16|13|14 15 16|13|14|15 * Pan-diagonal
Equations * .--.-----------.--.--. n1+ n6+n11+n16 = C ... pd1
n3|n4|n1|n2|n3|n4|n1|n2|n3 n1+ n8+n11+n14 = C ... pd2
.--|--+--+--+--|--. -| n2+ n7+n12+n13 = C ... pd3
n7|n8|n5|n6|n7|n8|n5|n6|n7 n2+ n5+n12+n15 = C ... pd4 |-
|--+--+--+--| -| -| n3+ n8+ n9+n14 = C ... pd5
11|12|n9|10|11|12|n9|10|11 n3+ n6+ n9+n16 = C ... pd6 |-
|--+--+--+--| -' -| n4+ n5+n10+n15 = C ... pd7
15|16|13|14|15|16|13|14|15 n4+ n7+n10+n13 = C ... pd8
'--'-----------'--'--' (C means the magic constant 34.) n3 n4|n1|n2
n3 n4|n1|n2 n3 '--'--------'--'
The lower figure above shows the Extended Space and the moving
frame of 4x4 square. This diagram is very useful for us to find any
pan-diagonals, because we can always see them in 'straight lines'
instead of those notorious 'broken, wrapping lines.' Pan-diagonal
Equations pd1~pd8 can be easily taken out of the Extended
Space.
You can move any last row or column of the supposed square far
to the opposite side and put it beyond the first, and you can make
a new Pan-magic square, as you know. In the Extended Space you can
realize the same thing by moving the 'square frame' as illustrated
above. New squares can be taken out only by picking up any 4x4
entries with the new frames moved to anywhere else. * 4x4 New
Squares Taken out of the Extended Space * 00/ 01/ 02/ 03/ n1 n2 n3
n4 13 14 15 16 9 10 11 12 5 6 7 8 n5 n6 n7 n8 1 2 3 4 13 14 15 16 9
10 11 12 n9 10 11 12 5 6 7 8 1 2 3 4 13 14 15 16 13 14 15 16 9 10
11 12 5 6 7 8 1 2 3 4
-
5
10/ 11/ 12/ 13/ 4 1 2 3 16 13 14 15 12 9 10 11 8 5 6 7 8 5 6 7 4
1 2 3 16 13 14 15 12 9 10 11 12 9 10 11 8 5 6 7 4 1 2 3 16 13 14 15
16 13 14 15 12 9 10 11 8 5 6 7 4 1 2 3
20/ 21/ 22/ 23/ 3 4 1 2 15 16 13 14 11 12 9 10 7 8 5 6 7 8 5 6 3
4 1 2 15 16 13 14 11 12 9 10 11 12 9 10 7 8 5 6 3 4 1 2 15 16 13 14
15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2 30/ 31/ 32/ 33/ 2 3 4 1 14
15 16 13 10 11 12 9 6 7 8 5 6 7 8 5 2 3 4 1 14 15 16 13 10 11 12 9
10 11 12 9 6 7 8 5 2 3 4 1 14 15 16 13 14 15 16 13 10 11 12 9 6 7 8
5 2 3 4 1
You can finally take out those new Pan-magic squares above as
many as 4x4. In any form two diagonals must add up to the magic
constant C(=34), since it should just satisfy the original concept
of 'pan-magic square' of order 4. The constant sum of every
pandiagonal makes it possible. #2. How many Solutions can we find?
What does it actually look like? Now let's make our computer draw
and count all pan-magic squares 4x4 at first, shall we? We know
there are three levels of solutions as follows: (1) 'Primitive',
(2) 'Standard', and (3) 'Fundamental'. Let's have them all here. **
384 'Primitive Solutions' of Pan-Diagonal Magic Squares 4x4 ** 1/
2/ 3/ 4/ 5/ 6/ 1 8 10 15 1 8 10 15 1 8 11 14 1 8 11 14 1 8 13 12 1
8 13 12 12 13 3 6 14 11 5 4 12 13 2 7 15 10 5 4 14 11 2 7 15 10 3 6
7 2 16 9 7 2 16 9 6 3 16 9 6 3 16 9 4 5 16 9 4 5 16 9 14 11 5 4 12
13 3 6 15 10 5 4 12 13 2 7 15 10 3 6 14 11 2 7 7/ 8/ 9/ 10/ 11/ 12/
1 12 6 15 1 12 6 15 1 12 7 14 1 12 7 14 1 12 13 8 1 12 13 8 8 13 3
10 14 7 9 4 8 13 2 11 15 6 9 4 14 7 2 11 15 6 3 10 11 2 16 5 11 2
16 5 10 3 16 5 10 3 16 5 4 9 16 5 4 9 16 5 14 7 9 4 8 13 3 10 15 6
9 4 8 13 2 11 15 6 3 10 14 7 2 11 13/ 14/ 15/ 16/ 17/ 18/ 1 14 4 15
1 14 4 15 1 14 7 12 1 14 7 12 1 14 11 8 1 14 11 8 8 11 5 10 12 7 9
6 8 11 2 13 15 4 9 6 12 7 2 13 15 4 5 10 13 2 16 3 13 2 16 3 10 5
16 3 10 5 16 3 6 9 16 3 6 9 16 3 12 7 9 6 8 11 5 10 15 4 9 6 8 11 2
13 15 4 5 10 12 7 2 13 19/ 20/ 21/ 22/ 23/ 24/ 1 15 4 14 1 15 4 14
1 15 6 12 1 15 6 12 1 15 10 8 1 15 10 8 8 10 5 11 12 6 9 7 8 10 3
13 14 4 9 7 12 6 3 13 14 4 5 11 13 3 16 2 13 3 16 2 11 5 16 2 11 5
16 2 7 9 16 2 7 9 16 2 12 6 9 7 8 10 5 11 14 4 9 7 8 10 3 13 14 4 5
11 12 6 3 13 25/ 26/ 27/ 28/ 29/ 30/ 2 7 9 16 2 7 9 16 2 7 12 13 2
7 12 13 2 7 14 11 2 7 14 11 11 14 4 5 13 12 6 3 11 14 1 8 16 9 6 3
13 12 1 8 16 9 4 5 8 1 15 10 8 1 15 10 5 4 15 10 5 4 15 10 3 6 15
10 3 6 15 10 13 12 6 3 11 14 4 5 16 9 6 3 11 14 1 8 16 9 4 5 13 12
1 8
-
6
31/ 32/ 33/ 34/ 35/ 36/ 2 11 5 16 2 11 5 16 2 11 8 13 2 11 8 13
2 11 14 7 2 11 14 7 7 14 4 9 13 8 10 3 7 14 1 12 16 5 10 3 13 8 1
12 16 5 4 9 12 1 15 6 12 1 15 6 9 4 15 6 9 4 15 6 3 10 15 6 3 10 15
6 13 8 10 3 7 14 4 9 16 5 10 3 7 14 1 12 16 5 4 9 13 8 1 12 37/ 38/
39/ 40/ 41/ 42/ 2 13 3 16 2 13 3 16 2 13 8 11 2 13 8 11 2 13 12 7 2
13 12 7 7 12 6 9 11 8 10 5 7 12 1 14 16 3 10 5 11 8 1 14 16 3 6 9
14 1 15 4 14 1 15 4 9 6 15 4 9 6 15 4 5 10 15 4 5 10 15 4 11 8 10 5
7 12 6 9 16 3 10 5 7 12 1 14 16 3 6 9 11 8 1 14 43/ 44/ 45/ 46/ 47/
48/ 2 16 3 13 2 16 3 13 2 16 5 11 2 16 5 11 2 16 9 7 2 16 9 7 7 9 6
12 11 5 10 8 7 9 4 14 13 3 10 8 11 5 4 14 13 3 6 12 14 4 15 1 14 4
15 1 12 6 15 1 12 6 15 1 8 10 15 1 8 10 15 1 11 5 10 8 7 9 6 12 13
3 10 8 7 9 4 14 13 3 6 12 11 5 4 14 49/ 50/ 51/ 52/ 53/ 54/ 3 6 9
16 3 6 9 16 3 6 12 13 3 6 12 13 3 6 15 10 3 6 15 10 10 15 4 5 13 12
7 2 10 15 1 8 16 9 7 2 13 12 1 8 16 9 4 5 8 1 14 11 8 1 14 11 5 4
14 11 5 4 14 11 2 7 14 11 2 7 14 11 13 12 7 2 10 15 4 5 16 9 7 2 10
15 1 8 16 9 4 5 13 12 1 8 55/ 56/ 57/ 58/ 59/ 60/ 3 10 5 16 3 10 5
16 3 10 8 13 3 10 8 13 3 10 15 6 3 10 15 6 6 15 4 9 13 8 11 2 6 15
1 12 16 5 11 2 13 8 1 12 16 5 4 9 12 1 14 7 12 1 14 7 9 4 14 7 9 4
14 7 2 11 14 7 2 11 14 7 13 8 11 2 6 15 4 9 16 5 11 2 6 15 1 12 16
5 4 9 13 8 1 12 61/ 62/ 63/ 64/ 65/ 66/ 3 13 2 16 3 13 2 16 3 13 8
10 3 13 8 10 3 13 12 6 3 13 12 6 6 12 7 9 10 8 11 5 6 12 1 15 16 2
11 5 10 8 1 15 16 2 7 9 15 1 14 4 15 1 14 4 9 7 14 4 9 7 14 4 5 11
14 4 5 11 14 4 10 8 11 5 6 12 7 9 16 2 11 5 6 12 1 15 16 2 7 9 10 8
1 15 67/ 68/ 69/ 70/ 71/ 72/ 3 16 2 13 3 16 2 13 3 16 5 10 3 16 5
10 3 16 9 6 3 16 9 6 6 9 7 12 10 5 11 8 6 9 4 15 13 2 11 8 10 5 4
15 13 2 7 12 15 4 14 1 15 4 14 1 12 7 14 1 12 7 14 1 8 11 14 1 8 11
14 1 10 5 11 8 6 9 7 12 13 2 11 8 6 9 4 15 13 2 7 12 10 5 4 15 73/
74/ 75/ 76/ 77/ 78/ 4 5 10 15 4 5 10 15 4 5 11 14 4 5 11 14 4 5 16
9 4 5 16 9 9 16 3 6 14 11 8 1 9 16 2 7 15 10 8 1 14 11 2 7 15 10 3
6 7 2 13 12 7 2 13 12 6 3 13 12 6 3 13 12 1 8 13 12 1 8 13 12 14 11
8 1 9 16 3 6 15 10 8 1 9 16 2 7 15 10 3 6 14 11 2 7 79/ 80/ 81/ 82/
83/ 84/ 4 9 6 15 4 9 6 15 4 9 7 14 4 9 7 14 4 9 16 5 4 9 16 5 5 16
3 10 14 7 12 1 5 16 2 11 15 6 12 1 14 7 2 11 15 6 3 10 11 2 13 8 11
2 13 8 10 3 13 8 10 3 13 8 1 12 13 8 1 12 13 8 14 7 12 1 5 16 3 10
15 6 12 1 5 16 2 11 15 6 3 10 14 7 2 11 85/ 86/ 87/ 88/ 89/ 90/ 4
14 1 15 4 14 1 15 4 14 7 9 4 14 7 9 4 14 11 5 4 14 11 5 5 11 8 10 9
7 12 6 5 11 2 16 15 1 12 6 9 7 2 16 15 1 8 10 16 2 13 3 16 2 13 3
10 8 13 3 10 8 13 3 6 12 13 3 6 12 13 3 9 7 12 6 5 11 8 10 15 1 12
6 5 11 2 16 15 1 8 10 9 7 2 16 91/ 92/ 93/ 94/ 95/ 96/ 4 15 1 14 4
15 1 14 4 15 6 9 4 15 6 9 4 15 10 5 4 15 10 5 5 10 8 11 9 6 12 7 5
10 3 16 14 1 12 7 9 6 3 16 14 1 8 11 16 3 13 2 16 3 13 2 11 8 13 2
11 8 13 2 7 12 13 2 7 12 13 2 9 6 12 7 5 10 8 11 14 1 12 7 5 10 3
16 14 1 8 11 9 6 3 16
-
7
97/ 98/ 99/ 100/ 101/ 102/ 5 4 9 16 5 4 9 16 5 4 14 11 5 4 14 11
5 4 15 10 5 4 15 10 10 15 6 3 11 14 7 2 10 15 1 8 16 9 7 2 11 14 1
8 16 9 6 3 8 1 12 13 8 1 12 13 3 6 12 13 3 6 12 13 2 7 12 13 2 7 12
13 11 14 7 2 10 15 6 3 16 9 7 2 10 15 1 8 16 9 6 3 11 14 1 8 . . .
. . . 355/ 356/ 357/ 358/ 359/ 360/ 15 10 3 6 15 10 3 6 15 10 5 4
15 10 5 4 15 10 8 1 15 10 8 1 1 8 13 12 4 5 16 9 1 8 11 14 6 3 16 9
4 5 11 14 6 3 13 12 14 11 2 7 14 11 2 7 12 13 2 7 12 13 2 7 9 16 2
7 9 16 2 7 4 5 16 9 1 8 13 12 6 3 16 9 1 8 11 14 6 3 13 12 4 5 11
14 361/ 362/ 363/ 364/ 365/ 366/ 16 2 7 9 16 2 7 9 16 2 11 5 16 2
11 5 16 2 13 3 16 2 13 3 3 13 12 6 5 11 14 4 3 13 8 10 9 7 14 4 5
11 8 10 9 7 12 6 10 8 1 15 10 8 1 15 6 12 1 15 6 12 1 15 4 14 1 15
4 14 1 15 5 11 14 4 3 13 12 6 9 7 14 4 3 13 8 10 9 7 12 6 5 11 8 10
367/ 368/ 369/ 370/ 371/ 372/ 16 3 6 9 16 3 6 9 16 3 10 5 16 3 10 5
16 3 13 2 16 3 13 2 2 13 12 7 5 10 15 4 2 13 8 11 9 6 15 4 5 10 8
11 9 6 12 7 11 8 1 14 11 8 1 14 7 12 1 14 7 12 1 14 4 15 1 14 4 15
1 14 5 10 15 4 2 13 12 7 9 6 15 4 2 13 8 11 9 6 12 7 5 10 8 11 373/
374/ 375/ 376/ 377/ 378/ 16 5 4 9 16 5 4 9 16 5 10 3 16 5 10 3 16 5
11 2 16 5 11 2 2 11 14 7 3 10 15 6 2 11 8 13 9 4 15 6 3 10 8 13 9 4
14 7 13 8 1 12 13 8 1 12 7 14 1 12 7 14 1 12 6 15 1 12 6 15 1 12 3
10 15 6 2 11 14 7 9 4 15 6 2 11 8 13 9 4 14 7 3 10 8 13 379/ 380/
381/ 382/ 383/ 384/ 16 9 4 5 16 9 4 5 16 9 6 3 16 9 6 3 16 9 7 2 16
9 7 2 2 7 14 11 3 6 15 10 2 7 12 13 5 4 15 10 3 6 12 13 5 4 14 11
13 12 1 8 13 12 1 8 11 14 1 8 11 14 1 8 10 15 1 8 10 15 1 8 3 6 15
10 2 7 14 11 5 4 15 10 2 7 12 13 5 4 14 11 3 6 12 13
* Counts according to the value of n1 * [ 1: 24] [ 2: 24] [ 3:
24] [ 4: 24] [ 5: 24] [ 6: 24] [ 7: 24] [ 8: 24] [ 9: 24] [10: 24]
[11: 24] [12: 24] [13: 24] [14: 24] [15: 24] [16: 24]
* Total Count = 384 * The next list is a sample program dictated
for the purpose of making the 'Standard Solutions' of pan-diagonal
magic squares of order 4. The list-forming inequality conditions I
put for them are {n1
-
8
void stp05(void), stp06(void), stp07(void), stp08(void); void
stp09(void), stp10(void), stp11(void), stp12(void); void
stp13(void), stp14(void), stp15(void), stp16(void); void
ansprint(void); void printans(short x); /**/ /* Main Program */ int
main(){ short n; printf("\n** 48 Standard Solutions of Pan-diagonal
Magic Squares 4x4 **\n"); for(n=0;n0){printans(cnt2);} printf("
[Count = %d]\n",cnt); printf(" OK!\n"); return 0; } /* Begin the
Calculations */ /* Set n1 */ void stp01(){ short a; for(a=1;a
-
9
/* Set n4 & n1
-
10
for(a=1;a
-
11
ansprint(); uflg[a]=0;}} } /**/ /* Print the Answers */ void
ansprint(){ short n; cnt++; anm[cnt2][0]=cnt; for(n=1;n
-
12
13/ 14/ 15/ 16/ 17/ 18/ .--.--.--.--. .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. .--.--.--.--. | 2|
7|12|13| | 2| 7| 9|16| | 2| 7|14|11| | 2| 7| 9|16| | 2| 7|14|11| |
2| 7|12|13| |--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |11|14| 1| 8| |11|14| 4| 5| |13|12| 1|
8| |13|12| 6| 3| |16| 9| 4| 5| |16| 9| 6| 3| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| | 5| 4|15|10| | 8| 1|15|10| | 3| 6|15|10| | 8|
1|15|10| | 3| 6|15|10| | 5| 4|15|10| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--| |16| 9| 6|
3| |13|12| 6| 3| |16| 9| 4| 5| |11|14| 4| 5| |13|12| 1| 8| |11|14|
1| 8| '--'--'--'--' '--'--'--'--' '--'--'--'--' '--'--'--'--'
'--'--'--'--' '--'--'--'--'
19/ 20/ 21/ 22/ 23/ 24/ .--.--.--.--. .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. .--.--.--.--. | 2|11|14|
7| | 2|11| 5|16| | 2|11|14| 7| | 2|11| 8|13| | 2|13|12| 7| | 2|13|
8|11| |--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |13| 8| 1|12| |13| 8|10| 3| |16| 5| 4|
9| |16| 5|10| 3| |16| 3| 6| 9| |16| 3|10| 5| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| | 3|10|15| 6| |12| 1|15| 6| | 3|10|15| 6| | 9| 4|15|
6| | 5|10|15| 4| | 9| 6|15| 4| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--| |16| 5| 4|
9| | 7|14| 4| 9| |13| 8| 1|12| | 7|14| 1|12| |11| 8| 1|14| | 7|12|
1|14| '--'--'--'--' '--'--'--'--' '--'--'--'--' '--'--'--'--'
'--'--'--'--' '--'--'--'--'
25/ 26/ 27/ 28/ 29/ 30/ .--.--.--.--. .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. .--.--.--.--. | 3|
6|15|10| | 3| 6| 9|16| | 3| 6|15|10| | 3| 6|12|13| | 3|10|15| 6| |
3|10| 5|16| |--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |13|12| 1| 8| |13|12| 7| 2| |16| 9| 4|
5| |16| 9| 7| 2| |13| 8| 1|12| |13| 8|11| 2| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| | 2| 7|14|11| | 8| 1|14|11| | 2| 7|14|11| | 5|
4|14|11| | 2|11|14| 7| |12| 1|14| 7| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--| |16| 9| 4|
5| |10|15| 4| 5| |13|12| 1| 8| |10|15| 1| 8| |16| 5| 4| 9| | 6|15|
4| 9| '--'--'--'--' '--'--'--'--' '--'--'--'--' '--'--'--'--'
'--'--'--'--' '--'--'--'--'
31/ 32/ 33/ 34/ 35/ 36/ .--.--.--.--. .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. .--.--.--.--. | 3|10|15|
6| | 3|10| 8|13| | 3|13|12| 6| | 3|13| 8|10| | 4| 5|16| 9| | 4|
5|10|15| |--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |16| 5| 4| 9| |16| 5|11| 2| |16| 2| 7|
9| |16| 2|11| 5| |14|11| 2| 7| |14|11| 8| 1| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| | 2|11|14| 7| | 9| 4|14| 7| | 5|11|14| 4| | 9| 7|14|
4| | 1| 8|13|12| | 7| 2|13|12| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--| |13| 8|
1|12| | 6|15| 1|12| |10| 8| 1|15| | 6|12| 1|15| |15|10| 3| 6| |
9|16| 3| 6| '--'--'--'--' '--'--'--'--' '--'--'--'--' '--'--'--'--'
'--'--'--'--' '--'--'--'--'
37/ 38/ 39/ 40/ 41/ 42/ .--.--.--.--. .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. .--.--.--.--. | 4| 5|16|
9| | 4| 5|11|14| | 4| 9|16| 5| | 4| 9| 6|15| | 4| 9|16| 5| | 4| 9|
7|14| |--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |15|10| 3| 6| |15|10| 8| 1| |14| 7|
2|11| |14| 7|12| 1| |15| 6| 3|10| |15| 6|12| 1| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| | 1| 8|13|12| | 6| 3|13|12| | 1|12|13| 8| |11| 2|13|
8| | 1|12|13| 8| |10| 3|13| 8| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--| |14|11| 2|
7| | 9|16| 2| 7| |15| 6| 3|10| | 5|16| 3|10| |14| 7| 2|11| | 5|16|
2|11| '--'--'--'--' '--'--'--'--' '--'--'--'--' '--'--'--'--'
'--'--'--'--' '--'--'--'--'
43/ 44/ 45/ 46/ 47/ 48/ .--.--.--.--. .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. .--.--.--.--. | 4|14|11|
5| | 4|14| 7| 9| | 5| 4|15|10| | 5| 4|14|11| | 6| 3|16| 9| | 6|
3|13|12| |--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |15| 1| 8|10| |15| 1|12| 6| |16| 9| 6|
3| |16| 9| 7| 2| |15|10| 5| 4| |15|10| 8| 1| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| | 6|12|13| 3| |10| 8|13| 3| | 2| 7|12|13| | 3|
6|12|13| | 1| 8|11|14| | 4| 5|11|14| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--| | 9| 7|
2|16| | 5|11| 2|16| |11|14| 1| 8| |10|15| 1| 8| |12|13| 2| 7| |
9|16| 2| 7| '--'--'--'--' '--'--'--'--' '--'--'--'--' '--'--'--'--'
'--'--'--'--' '--'--'--'--'
[Count = 48] OK! It may be of this type what you often see for
the solution list of our object. But it really changes its 'faces'
according to the list forming inequality conditions taken. The next
list is made up under the conditions: n1==1; n2
-
13
** 'Fundamental Three' of Pan-diagonal Magic Squares 4x4 ** [1]
5 4 15 10 5 4 15 10 .--.--.--.--. * Sums of Rows and Columns * 11
14| 1| 8|11|14| 1 8 1+ 8+11+14 = 34 ... rw1 |--+--+--+--| 12+13+ 2+
7 = 34 ... rw2 2 7|12|13| 2| 7|12 13 6+ 3+16+ 9 = 34 ... rw3
|--+--+--+--| 15+10+ 5+ 4 = 34 ... rw4 16 9| 6| 3|16| 9| 6 3 1+12+
6+15 = 34 ... cl1 |--+--+--+--| 8+13+ 3+10 = 34 ... cl2 5 4|15|10|
5| 4|15 10 11+ 2+16+ 5 = 34 ... cl3 '--'--'--'--' 14+ 7+ 9+ 4 = 34
... cl4 11 14 1 8 11 14 1 8 * Sums of Pan-diagonals * 1+13+16+ 4 =
34 ... pd1; 1+ 7+16+10 = 34 ... pb1; 8+ 2+ 9+15 = 34 ... pd2; 8+12+
9+ 5 = 34 ... pb2; 11+ 7+ 6+10 = 34 ... pd3; 11+13+ 6+ 4 = 34 ...
pb3; 14+12+ 3+ 5 = 34 ... pd4; 14+ 2+ 3+15 = 34 ... pb4; ... This
is certainly a Pan-diagonal Magic Type!
[2] 5 4 14 11 5 4 14 11 .--.--.--.--. * Sums of Rows and Columns
* 10 15| 1| 8|10|15| 1 8 1+ 8+10+15 = 34 ... rw1 |--+--+--+--|
12+13+ 3+ 6 = 34 ... rw2 3 6|12|13| 3| 6|12 13 7+ 2+16+ 9 = 34 ...
rw3 |--+--+--+--| 14+11+ 5+ 4 = 34 ... rw4 16 9| 7| 2|16| 9| 7 2
1+12+ 7+14 = 34 ... cl1 |--+--+--+--| 8+13+ 2+11 = 34 ... cl2 5
4|14|11| 5| 4|14 11 10+ 3+16+ 5 = 34 ... cl3 '--'--'--'--' 15+ 6+
9+ 4 = 34 ... cl4 10 15 1 8 10 15 1 8 * Sums of Pan-diagonals *
1+13+16+ 4 = 34 ... pd1; 1+ 6+16+11 = 34 ... pb1; 8+ 3+ 9+14 = 34
... pd2; 8+12+ 9+ 5 = 34 ... pb2; 10+ 6+ 7+11 = 34 ... pd3; 10+13+
7+ 4 = 34 ... pb3; 15+12+ 2+ 5 = 34 ... pd4; 15+ 3+ 2+14 = 34 ...
pb4; ... This is certainly a Pan-diagonal Magic Type!
[3] 3 6 15 10 3 6 15 10 .--.--.--.--. * Sums of Rows and Columns
* 13 12| 1| 8|13|12| 1 8 1+ 8+13+12 = 34 ... rw1 |--+--+--+--|
14+11+ 2+ 7 = 34 ... rw2 2 7|14|11| 2| 7|14 11 4+ 5+16+ 9 = 34 ...
rw3 |--+--+--+--| 15+10+ 3+ 6 = 34 ... rw4 16 9| 4| 5|16| 9| 4 5
1+14+ 4+15 = 34 ... cl1 |--+--+--+--| 8+11+ 5+10 = 34 ... cl2 3
6|15|10| 3| 6|15 10 13+ 2+16+ 3 = 34 ... cl3 '--'--'--'--' 12+ 7+
9+ 6 = 34 ... cl4 13 12 1 8 13 12 1 8 * Sums of Pan-diagonals *
1+11+16+ 6 = 34 ... pd1; 1+ 7+16+10 = 34 ... pb1; 8+ 2+ 9+15 = 34
... pd2; 8+14+ 9+ 3 = 34 ... pb2; 13+ 7+ 4+10 = 34 ... pd3; 13+11+
4+ 6 = 34 ... pb3; 12+14+ 5+ 3 = 34 ... pd4; 12+ 2+ 5+15 = 34 ...
pb4; ... This is certainly a Pan-diagonal Magic Type!
-
14
[Count = 3] OK! #3. Various Types of Pan-magic Squares have the
Same Solution Set. Can you imagine what solution sets you might
make for pan-magic squares 4x4 under such four different
definitions as listed below? No.1 is just the same with the
previous definition of this section, as you see. In the case of
order 8, we know we could have four independent sets of solutions,
different from each other, under similar style of definitions. How
about order 4? 1. [Basic Form and Basic Definitions] for
'Pan-diagonal' Type: 15 16 13 14 15 16 13 14 .--.--.--.--. * Rows
and Columns * 3 4|n1|n2|n3|n4| 1 2 n1+ n2+ n3+ n4 = C ... rw1
|--+--+--+--| n5+ n6+ n7+ n8 = C ... rw2 7 8|n5|n6|n7|n8| 5 6
n9+n10+n11+n12 = C ... rw3 |--+--+--+--| n13+n14+n15+n16 = C ...
rw4 11 12|n9|10|11|12| 9 10 n1+ n5+ n9+n13 = C ... cl1
|--+--+--+--| n2+ n6+n10+n14 = C ... cl2 15 16|13|14|15|16|13 14
n3+ n7+n11+n15 = C ... cl3 '--'--'--'--' n4+ n8+n12+n16 = C ... cl4
3 4 1 2 3 4 1 2 * Pan-diagonal Conditions * n1+ n6+n11+n16 = C ...
pd1; n1+ n8+n11+n14 = C ... pb1; n2+ n7+n12+n13 = C ... pd2; n2+
n5+n12+n15 = C ... pb2; n3+ n8+ n9+n14 = C ... pd3; n3+ n6+ n9+n16
= C ... pb3; n4+ n5+n10+n15 = C ... pd4; n4+ n7+n10+n13 = C ...
pb4; 2. [Basic Form and Basic Definitions] for 'Complete' Type:
.--.--.--.--. * Rows and Columns * |n1|n2|n3|n4| n1+ n2+ n3+ n4 = C
... rw1 |--+--+--+--| n5+ n6+ n7+ n8 = C ... rw2 |n5|n6|n7|n8|
n9+n10+n11+n12 = C ... rw3 |--+--+--+--| n13+n14+n15+n16 = C ...
rw4 |n9|10|11|12| n1+ n5+ n9+n13 = C ... cl1 |--+--+--+--| n2+
n6+n10+n14 = C ... cl2 |13|14|15|16| n3+ n7+n11+n15 = C ... cl3
'--'--'--'--' n4+ n8+n12+n16 = C ... cl4 * Complete Conditions *
n1+n11=n2+n12=n3+n9=n4+n10=n5+n15=n6+n16=n7+n13=n8+n14=CC=C/2 ...
cmplt 3. [Basic Form and Basic Definitions] for 'Composite' Type:
.--.--.--.--. * Composite Conditions * |n1|n2|n3|n4| n1+ n2+ n5+ n6
= C ... cst1 |--+--+--+--| n2+ n3+ n6+ n7 = C ... cst2
|n5|n6|n7|n8| n3+ n4+ n7+ n8 = C ... cst3 |--+--+--+--| n5+ n6+
n9+n10 = C ... cst4 |n9|10|11|12| n6+ n7+n10+n11 = C ... cst5
|--+--+--+--| n7+ n8+n11+n12 = C ... cst6 |13|14|15|16|
n9+n10+n13+n14 = C ... cst7 '--'--'--'--' n10+n11+n14+n15 = C ...
cst8 n11+n12+n15+n16 = C ... cst9 * Additional Conditions *
n1+n2+n3+n4=C ...rw1; n1+n5+n9+n13=C ...cl1; n1+n6+n11+n16=C
...pd1;
-
15
4. [Basic Form and Basic Definitions] for 'Composite and
Complete' Type: .--.--.--.--. * Composite Conditions * * Row and
Column * |n1|n2|n3|n4| n1+ n2+ n5+ n6 = C ...cst1 n1+ n2+ n3+ n4 =
C ...rw1 |--+--+--+--| n2+ n3+ n6+ n7 = C ...cst2 n1+ n5+ n9+n13 =
C ...cl1 |n5|n6|n7|n8| n3+ n4+ n7+ n8 = C ...cst3 |--+--+--+--| n5+
n6+ n9+n10 = C ...cst4 |n9|10|11|12| n6+ n7+n10+n11 = C ...cst5
|--+--+--+--| n7+ n8+n11+n12 = C ...cst6 |13|14|15|16|
n9+n10+n13+n14 = C ...cst7 '--'--'--'--' n10+n11+n14+n15 = C
...cst8 n11+n12+n15+n16 = C ...cst9 * Complete Conditions *
n1+n11=n2+n12=n3+n9=n4+n10=n5+n15=n6+n16=n7+n13=n8+n14=CC=C/2 ...
cmplt
In the case of order 4, whenever we start making under any of
those four different definitions above, we cannot but make the only
one solution set, completely the same in three levels. We cannot
make any other solution set at all. It is truly amazing. Every time
when I saw the same result of those four types, I used to be
extremely surprised. I began to think about what it really
means.
Let's have some algebraic study of our object here for a while.
#4. Important Properties of Pan-Diagonal Magic Squares I want to
start our argument with the Definition No.1 for 'Pan-diagonal'
type. 15 16 13 14 15 16 13 14 .--.--.--.--. * Rows and Columns * 3
4|n1|n2|n3|n4| 1 2 n1+ n2+ n3+ n4 = C ... rw1 |--+--+--+--| n5+ n6+
n7+ n8 = C ... rw2 7 8|n5|n6|n7|n8| 5 6 n9+n10+n11+n12 = C ... rw3
|--+--+--+--| n13+n14+n15+n16 = C ... rw4 11 12|n9|10|11|12| 9 10
n1+ n5+ n9+n13 = C ... cl1 |--+--+--+--| n2+ n6+n10+n14 = C ... cl2
15 16|13|14|15|16|13 14 n3+ n7+n11+n15 = C ... cl3 '--'--'--'--'
n4+ n8+n12+n16 = C ... cl4 3 4 1 2 3 4 1 2 n1+ n6+n11+n16 = C ...
pd1; n1+ n8+n11+n14 = C ... pb1; n2+ n7+n12+n13 = C ... pd2; n2+
n5+n12+n15 = C ... pb2; n3+ n8+ n9+n14 = C ... pd3; n3+ n6+ n9+n16
= C ... pb3; n4+ n5+n10+n15 = C ... pd4; n4+ n7+n10+n13 = C ...
pb4;
rw1+rw2+rw3+rw4 n1+n2+n3+n4+n5+n6+n7+n8+n9+ ... +n15+n16 = 4*C
1+2+3+4+5+6+7+8+9+ ... +15+16 = 4*C (1+16)*16/2 = 4*C Therefore C =
34 ... r0 Compare and Calculate the Pandiagonal equations as
follows: n1+ n6+n11+n16=C ... pd1 --> (n1+n11)+(n6+n16)=C n1+
n8+n11+n14=C ... pb1 --> (n1+n11)+(n8+n14)=C Therefore
n6+n16=n8+n14 (=CC1) n2+ n7+n12+n13=C ... pd2 -->
(n2+n12)+(n7+n13)=C n2+ n5+n12+n15=C ... pb2 -->
(n2+n12)+(n5+n15)=C Therefore n5+n15=n7+n13 (=CC2) n3+ n8+ n9+n14=C
... pd3 --> (n3+n9)+(n8+n14)=C
-
16
n3+ n6+ n9+n16=C ... pb3 --> (n3+n9)+(n6+n16)=C Therefore
n6+n16=n8+n14 (=CC1) n4+ n5+n10+n15=C ... pd4 -->
(n4+n10)+(n5+n15)=C n4+ n7+n10+n13=C ... pb4 -->
(n4+n10)+(n7+n13)=C Therefore n5+n15=n7+n13 (=CC2)
(n5+n15)+(n6+n16)+(n7+n13)+(n8+n14)=CC2+CC1+CC2+CC1
=(n5+n6+n7+n8)+(n13+n14+n15+n16)=2*C 2*(CC1+CC2)=2*C Therefore
CC1+CC2=C ... r1 In the same way (n1+n11)+(n6+n16)=C ... pd1
(n3+n9)+(n6+n16)=C ... pb3 --> n1+n11=n3+n9 (=DD1)
(n2+n12)+(n7+n13)=C ... pd2 (n4+n10)+(n7+n13)=C ... pb4 -->
n2+n12=n4+n10 (=DD2) (n1+n11)+(n2+n12)+(n3+n9)+(n4+n10)
=(n1+n2+n3+n4)+(n9+n10+n11+n12)=2*C 2*(DD1+DD2)=2*C Therefore
DD1+DD2=C ... r2 On the other hand (n1+n6+n11+n16)=C ... pd1
(n1+n11)+(n6+n16)=C; Therefore DD1+CC1=C ... r3 (n4+n7+n10+n13)=C
... pb4 (n4+n10)+(n7+n13)=C Therefore DD2+CC2=C ... r4 r1~r4 tell
us as a consequence: CC1=CC2=DD1=DD2=CC2=C/2=17 That means:
n1+n11=n2+n12=n3+n9=n4+n10 =n5+n16=n6+n16=n7+n13=n8+n14=17 ...
cmplt 8 Complementary Pairs of 17 are all located only on the
pan-diagonals. Each Pan-diagonal magic square 4x4 is always of
'Complete' type at the same time. #5. Pan-Magic Square and
Complementary Pairs The next figure shows where we can easily find
Complementary Pairs by the new notation. n1+n1=CC; n2+n2=CC;
n3+n3=CC; ... ; n7+n7=CC; n8+n8=CC (=17) We can also say in our
pan-diagonal object: n1=CC-n1=n11; n2=CC-n2=n12; n3=CC-n3=n9; ...;
n7=CC-n7=n13; n8=CC-n8=n14
[Complementary Pairs and Pan-Diagonals]
.-----------. n1+ n6+ n1+ n6 = C ...pd1' n3 n4|n1|n2|n3|n4|n1 n2
n3 n1+ n8+ n1+ n8 = C ...pb1' |--+--+--+--| n2+ n7+ n2+ n7 = C
...pd2' n7 n8|n5|n6|n7|n8|n5 n6 n7 n2+ n5+ n2+ n5 = C ...pb2'
|--+--+--+--| n3+ n8+ n3+ n8 = C ...pd3' n1 n2|n3|n4|n1|n2|n3 n4 n1
n3+ n6+ n3+ n6 = C ...pb3' |--+--+--+--| n4+ n5+ n4+ n5 = C ...pd4'
n5 n6|n7|n8|n5|n6|n7 n8 n5 n4+ n7+ n4+ n7 = C ...pb4'
'-----------'
Each pandiagonal consists of 2 complementary pairs of 17. Let's
put their own names for the four blocks in the next diagram as
shown below and calculate all sums of them.
{n1, n2, n5, n6}=block A; {n3, n4, n7, n8}=block B; {n3, n4, n7,
n8}=block C; {n1, n2, n5, n6}=block D;
-
17
.-----------. |n1|n2|n3|n4|
bA+bB=(n1+n2+n3+n4)+(n5+n6+n7+n8)=2*C; |-bA--|--bB-|
bA+bD=(n1+n1)+(n2+n2)+(n5+n5)+(n6+n6)=4x17=68=2*C; |n5|n6|n7|n8|
bB+bD=(n3+n7+n11+n15)+(n4+n8+n12+n16)=2*C; |-----+-----|
bB+bC=(n3+n3)+(n4+n4)+(n7+n7)+(n8+n8)=4x17=68=2*C; |n3|n4|n1|n2|
The first two shows bB=bD=2*C/2=C (=34) |-bC--|--bD-| The second
two shows bA=bB=2*C/2=C (=34) |n7|n8|n5|n6| ..... '-----------'
Consequently bA=bB=bC=bD=C (=34)
This means half of 'Composite Conditions' in Definition No.4 are
always true. What about {n2, n3, n6, n7}? n1+n2+n3+n4=C; +)
n5+n6+n7+n8=C; ----------------------
(n2+n3+n6+n7)+(n1+n4+n5+n8)=2*C But (n2+n3+n6+n7)+(n1+n4+n5+n8)=2*C
Therefore n1+n4+n5+n8 = n1+n4+n5+n8 But (n1+n5+n4+n8)+(n4+n8+n1+n5)
=(n1+n1)+(n4+n4)+(n5+n5)+(n8+n8)=4x17=68=2*C As a consequence
n1+n4+n5+n8 = n1+n4+n5+n8 = C That means n2+n3+n6+n7 = C In the
same way all the other half of 'Composite Conditions' are always
true.
After all we can say any Pan-diagonal magic square proves to be
always the 'Composite' pan-magic type, and also the 'Composite and
Complete' type at the same time, in the case of order 4. This makes
it possible that every type has the same solution set. Only our
algebraic study could reveal this inner relations among them. #6.
What Transformation System can we Assume? Let's go on our algebraic
study and step to the transformation system. Which lines can we
exchange in our object? Won't you see the next diagrams using the
new notation for 'Complementary Pairs of 17'?
Let me skip all those boring explanations and jump over to the
conclusion. We know we can exchange the 2nd4th row/column with each
other, and also exchange the 1st3rd row/column with each other. We
could even combine those two types of exchanges without destroying
anything about the constant sums we always expect. Therefore, we
can take such a powerful transformation system with this
possibility of line exchanges as follows. [Possibility of Line
Exchange] Basic Form (ii-iv)Exchange (II-IV)Exchange .-----------.
.-----------. .-----------. I|n1|n2|n3|n4| I|n1|n4|n3|n2|
I|n1|n2|n3|n4| |--+--+--+--| |--+--+--+--| |--+--+--+--|
II|n5|n6|n7|n8| II|n5|n8|n7|n6| IV|n7|n8|n5|n6| |--+--+--+--|
|--+--+--+--| |--+--+--+--| III|n3|n4|n1|n2| III|n3|n2|n1|n4|
III|n3|n4|n1|n2| |--+--+--+--| |--+--+--+--| |--+--+--+--|
IV|n7|n8|n5|n6| IV|n7|n6|n5|n8| II|n5|n6|n7|n8| '-----------'
'-----------' '-----------' i ii iii iv i iv iii ii i ii iii iv
-
18
(ii-iv)&(II-IV)Ex (i-iii)&(I-III)Ex .-----------.
.-----------. I|n1|n4|n3|n2| III|n1|n4|n3|n2| |--+--+--+--|
|--+--+--+--| IV|n7|n6|n5|n8| II|n7|n6|n5|n8| |--+--+--+--|
|--+--+--+--| III|n3|n2|n1|n4| I|n3|n2|n1|n4| |--+--+--+--|
|--+--+--+--| II|n5|n8|n7|n6| IV|n5|n8|n7|n6| '-----------'
'-----------' i iv iii ii iii ii i iv
Please see the next diagram I presented at the beginning of this
section here again. * 4x4 New Squares Taken out of the Extended
Space * 00/ 01/ 02/ 03/ n1 n2 n3 n4 13 14 15 16 9 10 11 12 5 6 7 8
n5 n6 n7 n8 1 2 3 4 13 14 15 16 9 10 11 12 n9 10 11 12 5 6 7 8 1 2
3 4 13 14 15 16 13 14 15 16 9 10 11 12 5 6 7 8 1 2 3 4 10/ 11/ 12/
13/ 4 1 2 3 16 13 14 15 12 9 10 11 8 5 6 7 8 5 6 7 4 1 2 3 16 13 14
15 12 9 10 11 12 9 10 11 8 5 6 7 4 1 2 3 16 13 14 15 16 13 14 15 12
9 10 11 8 5 6 7 4 1 2 3 20/ 21/ 22/ 23/ 3 4 1 2 15 16 13 14 11 12 9
10 7 8 5 6 7 8 5 6 3 4 1 2 15 16 13 14 11 12 9 10 11 12 9 10 7 8 5
6 3 4 1 2 15 16 13 14 15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2 30/
31/ 32/ 33/ 2 3 4 1 14 15 16 13 10 11 12 9 6 7 8 5 6 7 8 5 2 3 4 1
14 15 16 13 10 11 12 9 10 11 12 9 6 7 8 5 2 3 4 1 14 15 16 13 14 15
16 13 10 11 12 9 6 7 8 5 2 3 4 1
We can explain what it really means by the line exchangeability,
can't we? Take 4 examples 00/,03/,30/ and 33/ and get them smart
with n1=1 always on the left top as shown below. What parts of
03/,30/ and 33/ are different from the ones of original 00/? Yes.
We can transform the original into others with those line
exchangeabilities between the 2nd4th rows/columns and the
combination of two types. 00/ 03/ 30/ 33/ .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. |n1|n2|n3|n4|
|n1|n2|n3|n4| |n1|n4|n3|n2| |n1|n4|n3|n2| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |n5|n6|n7|n8|
|13|14|15|16| |n5|n8|n7|n6| |13|16|15|14| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |n9|10|11|12|
|n9|10|11|12| |n9|12|11|10| |n9|12|11|10| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |13|14|15|16|
|n5|n6|n7|n8| |13|16|15|14| |n5|n8|n7|n6| '--'--'--'--'
'--'--'--'--' '--'--'--'--' '--'--'--'--' #7. Reconstruction of All
Pandiagonal MS44 from the Fundamental Three
-
19
Now let's have an experiment to reconstruct all 384 'primitive
solutions' from the 'Fundamental Three' by such transformation
system as listed below. * Transformation System for Pan-diagonal
MS44 * 1. [4x4 Transformations by the 'Row/Column Shift'] 00/ 01/
02/ 03/ 1 2 3 4 13 14 15 16 9 10 11 12 5 6 7 8 5 6 7 8 1 2 3 4 13
14 15 16 9 10 11 12 9 10 11 12 5 6 7 8 1 2 3 4 13 14 15 16 13 14 15
16 9 10 11 12 5 6 7 8 1 2 3 4
10/ 11/ 12/ 13/ 4 1 2 3 16 13 14 15 12 9 10 11 8 5 6 7 8 5 6 7 4
1 2 3 16 13 14 15 12 9 10 11 12 9 10 11 8 5 6 7 4 1 2 3 16 13 14 15
16 13 14 15 12 9 10 11 8 5 6 7 4 1 2 3
20/ 21/ 22/ 23/ 3 4 1 2 15 16 13 14 11 12 9 10 7 8 5 6 7 8 5 6 3
4 1 2 15 16 13 14 11 12 9 10 11 12 9 10 7 8 5 6 3 4 1 2 15 16 13 14
15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2
30/ 31/ 32/ 33/ 2 3 4 1 14 15 16 13 10 11 12 9 6 7 8 5 6 7 8 5 2
3 4 1 14 15 16 13 10 11 12 9 10 11 12 9 6 7 8 5 2 3 4 1 14 15 16 13
14 15 16 13 10 11 12 9 6 7 8 5 2 3 4 1 2. [Mirror Reflection and
Rotation by 90 Degrees Clockwise] 00/ 01/ 02/ 03/ 1 2 3 4 13 9 5 1
16 15 14 13 4 8 12 16 5 6 7 8 14 10 6 2 12 11 10 9 3 7 11 15 9 10
11 12 15 11 7 3 8 7 6 5 2 6 10 14 13 14 15 16 16 12 8 4 4 3 2 1 1 5
9 13
10/ 11/ 12/ 13/ 1 5 9 13 4 3 2 1 16 12 8 4 13 14 15 16 2 6 10 14
8 7 6 5 15 11 7 3 9 10 11 12 3 7 11 15 12 11 10 9 14 10 6 2 5 6 7 8
4 8 12 16 16 15 14 13 13 9 5 1 1 2 3 4 We plan to make each
Fundamental into 16x8 forms by the combination of these two types
of transformation, and finally make 3x16x8=384 'primitive
solutions'. First of all, let me show you our starting points: the
Fundamental Three. And then you see the long list demonstrating the
result of our reconstruction. ** Reconstruct 'Pan-diagonal' Magic
Squares 4x4 ** ** Start: The 'Fundamental' Three Solutions: **
1/ 2/ 3/ 1 8 10 15 1 8 11 14 1 8 13 12 12 13 3 6 12 13 2 7 14 11
2 7 7 2 16 9 6 3 16 9 4 5 16 9 14 11 5 4 15 10 5 4 15 10 3 6 **
Transform them into 'Primitive' Solutions and Compare them with the
Old List **
-
20
[Fundamental #1]
1/ 330/ 74/ 353/ 8/ 360/ 79/ 335/
1 8 10 15 14 7 12 1 4 5 11 14 15 6 9 4 1 12 7 14 15 10 8 1 4 9 6
15 14 11 5 4 12 13 3 6 11 2 13 8 9 16 2 7 10 3 16 5 8 13 2 11 6 3
13 12 5 16 3 10 7 2 16 9 7 2 16 9 5 16 3 10 6 3 13 12 8 13 2 11 10
3 16 5 9 16 2 7 11 2 13 8 12 13 3 6 14 11 5 4 4 9 6 15 15 10 8 1 1
12 7 14 15 6 9 4 4 5 11 14 14 7 12 1 1 8 10 15
332/ 157/ 213/ 92/ 317/ 77/ 204/ 148/
14 11 5 4 7 12 1 14 9 16 2 7 4 15 6 9 14 1 12 7 4 5 11 14 9 6 15
4 7 2 16 9 1 8 10 15 2 13 8 11 6 3 13 12 5 10 3 16 11 8 13 2 15 10
8 1 16 3 10 5 12 13 3 6 12 13 3 6 16 3 10 5 15 10 8 1 11 8 13 2 5
10 3 16 6 3 13 12 2 13 8 11 1 8 10 15 7 2 16 9 9 6 15 4 4 5 11 14
14 1 12 7 4 15 6 9 9 16 2 7 7 12 1 14 14 11 5 4
150/ 269/ 125/ 198/ 163/ 211/ 140/ 284/
7 2 16 9 12 1 14 7 6 3 13 12 9 4 15 6 7 14 1 12 9 16 2 7 6 15 4
9 12 13 3 6 14 11 5 4 13 8 11 2 15 10 8 1 16 5 10 3 2 11 8 13 4 5
11 14 3 10 5 16 1 8 10 15 1 8 10 15 3 10 5 16 4 5 11 14 2 11 8 13
16 5 10 3 15 10 8 1 13 8 11 2 14 11 5 4 12 13 3 6 6 15 4 9 9 16 2 7
7 14 1 12 9 4 15 6 6 3 13 12 12 1 14 7 7 2 16 9
287/ 14/ 358/ 127/ 282/ 122/ 347/ 3/
12 13 3 6 1 14 7 12 15 10 8 1 6 9 4 15 12 7 14 1 6 3 13 12 15 4
9 6 1 8 10 15 7 2 16 9 8 11 2 13 4 5 11 14 3 16 5 10 13 2 11 8 9 16
2 7 10 5 16 3 14 11 5 4 14 11 5 4 10 5 16 3 9 16 2 7 13 2 11 8 3 16
5 10 4 5 11 14 8 11 2 13 7 2 16 9 1 8 10 15 15 4 9 6 6 3 13 12 12 7
14 1 6 9 4 15 15 10 8 1 1 14 7 12 12 13 3 6
339/ 81/ 114/ 220/ 350/ 229/ 119/ 88/
15 1 8 10 4 9 6 15 5 11 14 4 10 3 16 5 15 6 9 4 10 8 1 15 5 16 3
10 4 14 11 5 6 12 13 3 14 7 12 1 16 2 7 9 8 13 2 11 1 12 7 14 3 13
12 6 11 2 13 8 9 7 2 16 9 7 2 16 11 2 13 8 3 13 12 6 1 12 7 14 8 13
2 11 16 2 7 9 14 7 12 1 6 12 13 3 4 14 11 5 5 16 3 10 10 8 1 15 15
6 9 4 10 3 16 5 5 11 14 4 4 9 6 15 15 1 8 10
90/ 202/ 361/ 105/ 95/ 112/ 368/ 207/
4 14 11 5 9 6 15 4 16 2 7 9 5 10 3 16 4 15 6 9 5 11 14 4 16 3 10
5 9 7 2 16 15 1 8 10 7 12 1 14 3 13 12 6 11 8 13 2 14 1 12 7 10 8 1
15 2 13 8 11 6 12 13 3 6 12 13 3 2 13 8 11 10 8 1 15 14 1 12 7 11 8
13 2 3 13 12 6 7 12 1 14 15 1 8 10 9 7 2 16 16 3 10 5 5 11 14 4 4
15 6 9 5 10 3 16 16 2 7 9 9 6 15 4 4 14 11 5
205/ 143/ 64/ 374/ 196/ 363/ 57/ 138/
9 7 2 16 6 15 4 9 3 13 12 6 16 5 10 3 9 4 15 6 16 2 7 9 3 10 5
16 6 12 13 3 4 14 11 5 12 1 14 7 10 8 1 15 2 11 8 13 7 14 1 12 5 11
14 4 13 8 11 2 15 1 8 10 15 1 8 10 13 8 11 2 5 11 14 4 7 14 1 12 2
11 8 13 10 8 1 15 12 1 14 7 4 14 11 5 6 12 13 3 3 10 5 16 16 2 7 9
9 4 15 6 16 5 10 3 3 13 12 6 6 15 4 9 9 7 2 16
136/ 344/ 231/ 71/ 129/ 66/ 226/ 337/
6 12 13 3 15 4 9 6 10 8 1 15 3 16 5 10 6 9 4 15 3 13 12 6 10 5
16 3 15 1 8 10 9 7 2 16 1 14 7 12 5 11 14 4 13 2 11 8 12 7 14 1 16
2 7 9 8 11 2 13 4 14 11 5 4 14 11 5 8 11 2 13 16 2 7 9 12 7 14 1 13
2 11 8 5 11 14 4 1 14 7 12 9 7 2 16 15 1 8 10 10 5 16 3 3 13 12 6 6
9 4 15 3 16 5 10 10 8 1 15 15 4 9 6 6 12 13 3
235/ 116/ 260/ 187/ 222/ 174/ 245/ 101/
10 15 1 8 5 16 3 10 11 14 4 5 8 13 2 11 10 3 16 5 8 1 15 10 11 2
13 8 5 4 14 11 3 6 12 13 4 9 6 15 2 7 9 16 1 12 7 14 15 6 9 4 13 12
6 3 14 7 12 1 16 9 7 2 16 9 7 2 14 7 12 1 13 12 6 3 15 6 9 4 1 12 7
14 2 7 9 16 4 9 6 15 3 6 12 13 5 4 14 11 11 2 13 8 8 1 15 10 10 3
16 5 8 13 2 11 11 14 4 5 5 16 3 10 10 15 1 8
98/ 371/ 27/ 258/ 103/ 263/ 38/ 382/
5 4 14 11 16 3 10 5 2 7 9 16 11 8 13 2 5 10 3 16 11 14 4 5 2 13
8 11 16 9 7 2 10 15 1 8 9 6 15 4 13 12 6 3 14 1 12 7 4 15 6 9 8 1
15 10 7 12 1 14 3 6 12 13 3 6 12 13 7 12 1 14 8 1 15 10 4 15 6 9 14
1 12 7 13 12 6 3 9 6 15 4 10 15 1 8 16 9 7 2 2 13 8 11 11 14 4 5 5
10 3 16 11 8 13 2 2 7 9 16 16 3 10 5 5 4 14 11
384/ 55/ 311/ 32/ 377/ 25/ 306/ 50/
16 9 7 2 3 10 5 16 13 12 6 3 2 11 8 13 16 5 10 3 2 7 9 16 13 8
11 2 3 6 12 13 5 4 14 11 6 15 4 9 8 1 15 10 7 14 1 12 9 4 15 6 11
14 4 5 12 1 14 7 10 15 1 8 10 15 1 8 12 1 14 7 11 14 4 5 9 4 15 6 7
14 1 12 8 1 15 10 6 15 4 9 5 4 14 11 3 6 12 13 13 8 11 2 2 7 9 16
16 5 10 3 2 11 8 13 13 12 6 3 3 10 5 16 16 9 7 2
53/ 228/ 172/ 293/ 68/ 308/ 181/ 237/
3 6 12 13 10 5 16 3 8 1 15 10 13 2 11 8 3 16 5 10 13 12 6 3 8 11
2 13 10 15 1 8 16 9 7 2 15 4 9 6 11 14 4 5 12 7 14 1 6 9 4 15 2 7 9
16 1 14 7 12 5 4 14 11 5 4 14 11 1 14 7 12 2 7 9 16 6 9 4 15 12 7
14 1 11 14 4 5 15 4 9 6 16 9 7 2 10 15 1 8 8 11 2 13 13 12 6 3 3 16
5 10 13 2 11 8 8 1 15 10 10 5 16 3 3 6 12 13
-
21
180/ 242/ 321/ 11/ 189/ 22/ 328/ 247/
8 10 15 1 11 2 13 8 14 4 5 11 1 12 7 14 8 13 2 11 1 15 10 8 14 7
12 1 11 5 4 14 13 3 6 12 5 16 3 10 7 9 16 2 15 6 9 4 10 3 16 5 12 6
3 13 4 9 6 15 2 16 9 7 2 16 9 7 4 9 6 15 12 6 3 13 10 3 16 5 15 6 9
4 7 9 16 2 5 16 3 10 13 3 6 12 11 5 4 14 14 7 12 1 1 15 10 8 8 13 2
11 1 12 7 14 14 4 5 11 11 2 13 8 8 10 15 1
249/ 41/ 154/ 314/ 256/ 319/ 159/ 48/
11 5 4 14 2 13 8 11 7 9 16 2 14 1 12 7 11 8 13 2 14 4 5 11 7 12
1 14 2 16 9 7 8 10 15 1 16 3 10 5 12 6 3 13 4 15 6 9 5 10 3 16 1 15
10 8 9 6 15 4 13 3 6 12 13 3 6 12 9 6 15 4 1 15 10 8 5 10 3 16 4 15
6 9 12 6 3 13 16 3 10 5 8 10 15 1 2 16 9 7 7 12 1 14 14 4 5 11 11 8
13 2 14 1 12 7 7 9 16 2 2 13 8 11 11 5 4 14
46/ 304/ 271/ 165/ 35/ 156/ 266/ 297/
2 16 9 7 13 8 11 2 12 6 3 13 7 14 1 12 2 11 8 13 7 9 16 2 12 1
14 7 13 3 6 12 11 5 4 14 3 10 5 16 1 15 10 8 9 4 15 6 16 5 10 3 14
4 5 11 6 15 4 9 8 10 15 1 8 10 15 1 6 15 4 9 14 4 5 11 16 5 10 3 9
4 15 6 1 15 10 8 3 10 5 16 11 5 4 14 13 3 6 12 12 1 14 7 7 9 16 2 2
11 8 13 7 14 1 12 12 6 3 13 13 8 11 2 2 16 9 7
295/ 183/ 24/ 280/ 290/ 273/ 17/ 178/
13 3 6 12 8 11 2 13 1 15 10 8 12 7 14 1 13 2 11 8 12 6 3 13 1 14
7 12 8 10 15 1 2 16 9 7 10 5 16 3 14 4 5 11 6 9 4 15 3 16 5 10 7 9
16 2 15 4 9 6 11 5 4 14 11 5 4 14 15 4 9 6 7 9 16 2 3 16 5 10 6 9 4
15 14 4 5 11 10 5 16 3 2 16 9 7 8 10 15 1 1 14 7 12 12 6 3 13 13 2
11 8 12 7 14 1 1 15 10 8 8 11 2 13 13 3 6 12
[Fundamental #2] 2/ 354/ 73/ 329/ 7/ 336/ 80/ 359/ 1 8 11 14 15
6 12 1 4 5 10 15 14 7 9 4 1 12 6 15 14 11 8 1 4 9 7 14 15 10 5 4 12
13 2 7 10 3 13 8 9 16 3 6 11 2 16 5 8 13 3 10 7 2 13 12 5 16 2 11 6
3 16 9 6 3 16 9 5 16 2 11 7 2 13 12 8 13 3 10 11 2 16 5 9 16 3 6 10
3 13 8 12 13 2 7 15 10 5 4 4 9 7 14 14 11 8 1 1 12 6 15 14 7 9 4 4
5 10 15 15 6 12 1 1 8 11 14
356/ 133/ 215/ 86/ 341/ 75/ 210/ 124/
15 10 5 4 6 12 1 15 9 16 3 6 4 14 7 9 15 1 12 6 4 5 10 15 9 7 14
4 6 3 16 9 1 8 11 14 3 13 8 10 7 2 13 12 5 11 2 16 10 8 13 3 14 11
8 1 16 2 11 5 12 13 2 7 12 13 2 7 16 2 11 5 14 11 8 1 10 8 13 3 5
11 2 16 7 2 13 12 3 13 8 10 1 8 11 14 6 3 16 9 9 7 14 4 4 5 10 15
15 1 12 6 4 14 7 9 9 16 3 6 6 12 1 15 15 10 5 4
126/ 270/ 149/ 197/ 139/ 212/ 164/ 283/
6 3 16 9 12 1 15 6 7 2 13 12 9 4 14 7 6 15 1 12 9 16 3 6 7 14 4
9 12 13 2 7 15 10 5 4 13 8 10 3 14 11 8 1 16 5 11 2 3 10 8 13 4 5
10 15 2 11 5 16 1 8 11 14 1 8 11 14 2 11 5 16 4 5 10 15 3 10 8 13
16 5 11 2 14 11 8 1 13 8 10 3 15 10 5 4 12 13 2 7 7 14 4 9 9 16 3 6
6 15 1 12 9 4 14 7 7 2 13 12 12 1 15 6 6 3 16 9
285/ 20/ 334/ 151/ 276/ 146/ 323/ 5/
12 13 2 7 1 15 6 12 14 11 8 1 7 9 4 14 12 6 15 1 7 2 13 12 14 4
9 7 1 8 11 14 6 3 16 9 8 10 3 13 4 5 10 15 2 16 5 11 13 3 10 8 9 16
3 6 11 5 16 2 15 10 5 4 15 10 5 4 11 5 16 2 9 16 3 6 13 3 10 8 2 16
5 11 4 5 10 15 8 10 3 13 6 3 16 9 1 8 11 14 14 4 9 7 7 2 13 12 12 6
15 1 7 9 4 14 14 11 8 1 1 15 6 12 12 13 2 7
315/ 83/ 108/ 244/ 326/ 253/ 117/ 94/
14 1 8 11 4 9 7 14 5 10 15 4 11 2 16 5 14 7 9 4 11 8 1 14 5 16 2
11 4 15 10 5 7 12 13 2 15 6 12 1 16 3 6 9 8 13 3 10 1 12 6 15 2 13
12 7 10 3 13 8 9 6 3 16 9 6 3 16 10 3 13 8 2 13 12 7 1 12 6 15 8 13
3 10 16 3 6 9 15 6 12 1 7 12 13 2 4 15 10 5 5 16 2 11 11 8 1 14 14
7 9 4 11 2 16 5 5 10 15 4 4 9 7 14 14 1 8 11
96/ 208/ 367/ 111/ 89/ 106/ 362/ 201/
4 15 10 5 9 7 14 4 16 3 6 9 5 11 2 16 4 14 7 9 5 10 15 4 16 2 11
5 9 6 3 16 14 1 8 11 6 12 1 15 2 13 12 7 10 8 13 3 15 1 12 6 11 8 1
14 3 13 8 10 7 12 13 2 7 12 13 2 3 13 8 10 11 8 1 14 15 1 12 6 10 8
13 3 2 13 12 7 6 12 1 15 14 1 8 11 9 6 3 16 16 2 11 5 5 10 15 4 4
14 7 9 5 11 2 16 16 3 6 9 9 7 14 4 4 15 10 5
199/ 167/ 40/ 376/ 194/ 369/ 33/ 162/
9 6 3 16 7 14 4 9 2 13 12 7 16 5 11 2 9 4 14 7 16 3 6 9 2 11 5
16 7 12 13 2 4 15 10 5 12 1 15 6 11 8 1 14 3 10 8 13 6 15 1 12 5 10
15 4 13 8 10 3 14 1 8 11 14 1 8 11 13 8 10 3 5 10 15 4 6 15 1 12 3
10 8 13 11 8 1 14 12 1 15 6 4 15 10 5 7 12 13 2 2 11 5 16 16 3 6 9
9 4 14 7 16 5 11 2 2 13 12 7 7 14 4 9 9 6 3 16
160/ 320/ 255/ 47/ 153/ 42/ 250/ 313/
7 12 13 2 14 4 9 7 11 8 1 14 2 16 5 11 7 9 4 14 2 13 12 7 11 5
16 2 14 1 8 11 9 6 3 16 1 15 6 12 5 10 15 4 13 3 10 8 12 6 15 1 16
3 6 9 8 10 3 13 4 15 10 5 4 15 10 5 8 10 3 13 16 3 6 9 12 6 15 1 13
3 10 8 5 10 15 4 1 15 6 12 9 6 3 16 14 1 8 11 11 5 16 2 2 13 12 7 7
9 4 14 2 16 5 11 11 8 1 14 14 4 9 7 7 12 13 2
-
22
259/ 115/ 236/ 188/ 246/ 173/ 221/ 102/
11 14 1 8 5 16 2 11 10 15 4 5 8 13 3 10 11 2 16 5 8 1 14 11 10 3
13 8 5 4 15 10 2 7 12 13 4 9 7 14 3 6 9 16 1 12 6 15 14 7 9 4 13 12
7 2 15 6 12 1 16 9 6 3 16 9 6 3 15 6 12 1 13 12 7 2 14 7 9 4 1 12 6
15 3 6 9 16 4 9 7 14 2 7 12 13 5 4 15 10 10 3 13 8 8 1 14 11 11 2
16 5 8 13 3 10 10 15 4 5 5 16 2 11 11 14 1 8
100/ 365/ 51/ 234/ 109/ 239/ 62/ 380/
5 4 15 10 16 2 11 5 3 6 9 16 10 8 13 3 5 11 2 16 10 15 4 5 3 13
8 10 16 9 6 3 11 14 1 8 9 7 14 4 13 12 7 2 15 1 12 6 4 14 7 9 8 1
14 11 6 12 1 15 2 7 12 13 2 7 12 13 6 12 1 15 8 1 14 11 4 14 7 9 15
1 12 6 13 12 7 2 9 7 14 4 11 14 1 8 16 9 6 3 3 13 8 10 10 15 4 5 5
11 2 16 10 8 13 3 3 6 9 16 16 2 11 5 5 4 15 10
383/ 31/ 312/ 56/ 378/ 49/ 305/ 26/
16 9 6 3 2 11 5 16 13 12 7 2 3 10 8 13 16 5 11 2 3 6 9 16 13 8
10 3 2 7 12 13 5 4 15 10 7 14 4 9 8 1 14 11 6 15 1 12 9 4 14 7 10
15 4 5 12 1 15 6 11 14 1 8 11 14 1 8 12 1 15 6 10 15 4 5 9 4 14 7 6
15 1 12 8 1 14 11 7 14 4 9 5 4 15 10 2 7 12 13 13 8 10 3 3 6 9 16
16 5 11 2 3 10 8 13 13 12 7 2 2 11 5 16 16 9 6 3
29/ 252/ 170/ 299/ 44/ 310/ 175/ 261/
2 7 12 13 11 5 16 2 8 1 14 11 13 3 10 8 2 16 5 11 13 12 7 2 8 10
3 13 11 14 1 8 16 9 6 3 14 4 9 7 10 15 4 5 12 6 15 1 7 9 4 14 3 6 9
16 1 15 6 12 5 4 15 10 5 4 15 10 1 15 6 12 3 6 9 16 7 9 4 14 12 6
15 1 10 15 4 5 14 4 9 7 16 9 6 3 11 14 1 8 8 10 3 13 13 12 7 2 2 16
5 11 13 3 10 8 8 1 14 11 11 5 16 2 2 7 12 13
186/ 218/ 345/ 9/ 191/ 16/ 352/ 223/
8 11 14 1 10 3 13 8 15 4 5 10 1 12 6 15 8 13 3 10 1 14 11 8 15 6
12 1 10 5 4 15 13 2 7 12 5 16 2 11 6 9 16 3 14 7 9 4 11 2 16 5 12 7
2 13 4 9 7 14 3 16 9 6 3 16 9 6 4 9 7 14 12 7 2 13 11 2 16 5 14 7 9
4 6 9 16 3 5 16 2 11 13 2 7 12 10 5 4 15 15 6 12 1 1 14 11 8 8 13 3
10 1 12 6 15 15 4 5 10 10 3 13 8 8 11 14 1
225/ 65/ 130/ 338/ 232/ 343/ 135/ 72/
10 5 4 15 3 13 8 10 6 9 16 3 15 1 12 6 10 8 13 3 15 4 5 10 6 12
1 15 3 16 9 6 8 11 14 1 16 2 11 5 12 7 2 13 4 14 7 9 5 11 2 16 1 14
11 8 9 7 14 4 13 2 7 12 13 2 7 12 9 7 14 4 1 14 11 8 5 11 2 16 4 14
7 9 12 7 2 13 16 2 11 5 8 11 14 1 3 16 9 6 6 12 1 15 15 4 5 10 10 8
13 3 15 1 12 6 6 9 16 3 3 13 8 10 10 5 4 15
70/ 302/ 277/ 141/ 59/ 132/ 268/ 291/
3 16 9 6 13 8 10 3 12 7 2 13 6 15 1 12 3 10 8 13 6 9 16 3 12 1
15 6 13 2 7 12 10 5 4 15 2 11 5 16 1 14 11 8 9 4 14 7 16 5 11 2 15
4 5 10 7 14 4 9 8 11 14 1 8 11 14 1 7 14 4 9 15 4 5 10 16 5 11 2 9
4 14 7 1 14 11 8 2 11 5 16 10 5 4 15 13 2 7 12 12 1 15 6 6 9 16 3 3
10 8 13 6 15 1 12 12 7 2 13 13 8 10 3 3 16 9 6
289/ 177/ 18/ 274/ 296/ 279/ 23/ 184/
13 2 7 12 8 10 3 13 1 14 11 8 12 6 15 1 13 3 10 8 12 7 2 13 1 15
6 12 8 11 14 1 3 16 9 6 11 5 16 2 15 4 5 10 7 9 4 14 2 16 5 11 6 9
16 3 14 4 9 7 10 5 4 15 10 5 4 15 14 4 9 7 6 9 16 3 2 16 5 11 7 9 4
14 15 4 5 10 11 5 16 2 3 16 9 6 8 11 14 1 1 15 6 12 12 7 2 13 13 3
10 8 12 6 15 1 1 14 11 8 8 10 3 13 13 2 7 12
[Fundamental #3] 4/ 348/ 121/ 281/ 13/ 288/ 128/ 357/ 1 8 13 12
15 4 14 1 6 3 10 15 12 7 9 6 1 14 4 15 12 13 8 1 6 9 7 12 15 10 3 6
14 11 2 7 10 5 11 8 9 16 5 4 13 2 16 3 8 11 5 10 7 2 11 14 3 16 2
13 4 5 16 9 4 5 16 9 3 16 2 13 7 2 11 14 8 11 5 10 13 2 16 3 9 16 5
4 10 5 11 8 14 11 2 7 15 10 3 6 6 9 7 12 12 13 8 1 1 14 4 15 12 7 9
6 6 3 10 15 15 4 14 1 1 8 13 12
355/ 85/ 216/ 134/ 342/ 123/ 209/ 76/
15 10 3 6 4 14 1 15 9 16 5 4 6 12 7 9 15 1 14 4 6 3 10 15 9 7 12
6 4 5 16 9 1 8 13 12 5 11 8 10 7 2 11 14 3 13 2 16 10 8 11 5 12 13
8 1 16 2 13 3 14 11 2 7 14 11 2 7 16 2 13 3 12 13 8 1 10 8 11 5 3
13 2 16 7 2 11 14 5 11 8 10 1 8 13 12 4 5 16 9 9 7 12 6 6 3 10 15
15 1 14 4 6 12 7 9 9 16 5 4 4 14 1 15 15 10 3 6
78/ 318/ 147/ 203/ 91/ 214/ 158/ 331/
4 5 16 9 14 1 15 4 7 2 11 14 9 6 12 7 4 15 1 14 9 16 5 4 7 12 6
9 14 11 2 7 15 10 3 6 11 8 10 5 12 13 8 1 16 3 13 2 5 10 8 11 6 3
10 15 2 13 3 16 1 8 13 12 1 8 13 12 2 13 3 16 6 3 10 15 5 10 8 11
16 3 13 2 12 13 8 1 11 8 10 5 15 10 3 6 14 11 2 7 7 12 6 9 9 16 5 4
4 15 1 14 9 6 12 7 7 2 11 14 14 1 15 4 4 5 16 9
333/ 19/ 286/ 152/ 324/ 145/ 275/ 6/
14 11 2 7 1 15 4 14 12 13 8 1 7 9 6 12 14 4 15 1 7 2 11 14 12 6
9 7 1 8 13 12 4 5 16 9 8 10 5 11 6 3 10 15 2 16 3 13 11 5 10 8 9 16
5 4 13 3 16 2 15 10 3 6 15 10 3 6 13 3 16 2 9 16 5 4 11 5 10 8 2 16
3 13 6 3 10 15 8 10 5 11 4 5 16 9 1 8 13 12 12 6 9 7 7 2 11 14 14 4
15 1 7 9 6 12 12 13 8 1 1 15 4 14 14 11 2 7
-
23
267/ 131/ 60/ 292/ 278/ 301/ 69/ 142/
12 1 8 13 6 9 7 12 3 10 15 6 13 2 16 3 12 7 9 6 13 8 1 12 3 16 2
13 6 15 10 3 7 14 11 2 15 4 14 1 16 5 4 9 8 11 5 10 1 14 4 15 2 11
14 7 10 5 11 8 9 4 5 16 9 4 5 16 10 5 11 8 2 11 14 7 1 14 4 15 8 11
5 10 16 5 4 9 15 4 14 1 7 14 11 2 6 15 10 3 3 16 2 13 13 8 1 12 12
7 9 6 13 2 16 3 3 10 15 6 6 9 7 12 12 1 8 13
144/ 206/ 373/ 63/ 137/ 58/ 364/ 195/
6 15 10 3 9 7 12 6 16 5 4 9 3 13 2 16 6 12 7 9 3 10 15 6 16 2 13
3 9 4 5 16 12 1 8 13 4 14 1 15 2 11 14 7 10 8 11 5 15 1 14 4 13 8 1
12 5 11 8 10 7 14 11 2 7 14 11 2 5 11 8 10 13 8 1 12 15 1 14 4 10 8
11 5 2 11 14 7 4 14 1 15 12 1 8 13 9 4 5 16 16 2 13 3 3 10 15 6 6
12 7 9 3 13 2 16 16 5 4 9 9 7 12 6 6 15 10 3
193/ 161/ 34/ 370/ 200/ 375/ 39/ 168/
9 4 5 16 7 12 6 9 2 11 14 7 16 3 13 2 9 6 12 7 16 5 4 9 2 13 3
16 7 14 11 2 6 15 10 3 14 1 15 4 13 8 1 12 5 10 8 11 4 15 1 14 3 10
15 6 11 8 10 5 12 1 8 13 12 1 8 13 11 8 10 5 3 10 15 6 4 15 1 14 5
10 8 11 13 8 1 12 14 1 15 4 6 15 10 3 7 14 11 2 2 13 3 16 16 5 4 9
9 6 12 7 16 3 13 2 2 11 14 7 7 12 6 9 9 4 5 16
166/ 272/ 303/ 45/ 155/ 36/ 298/ 265/
7 14 11 2 12 6 9 7 13 8 1 12 2 16 3 13 7 9 6 12 2 11 14 7 13 3
16 2 12 1 8 13 9 4 5 16 1 15 4 14 3 10 15 6 11 5 10 8 14 4 15 1 16
5 4 9 8 10 5 11 6 15 10 3 6 15 10 3 8 10 5 11 16 5 4 9 14 4 15 1 11
5 10 8 3 10 15 6 1 15 4 14 9 4 5 16 12 1 8 13 13 3 16 2 2 11 14 7 7
9 6 12 2 16 3 13 13 8 1 12 12 6 9 7 7 14 11 2
307/ 67/ 238/ 182/ 294/ 171/ 227/ 54/
13 12 1 8 3 16 2 13 10 15 6 3 8 11 5 10 13 2 16 3 8 1 12 13 10 5
11 8 3 6 15 10 2 7 14 11 6 9 7 12 5 4 9 16 1 14 4 15 12 7 9 6 11 14
7 2 15 4 14 1 16 9 4 5 16 9 4 5 15 4 14 1 11 14 7 2 12 7 9 6 1 14 4
15 5 4 9 16 6 9 7 12 2 7 14 11 3 6 15 10 10 5 11 8 8 1 12 13 13 2
16 3 8 11 5 10 10 15 6 3 3 16 2 13 13 12 1 8
52/ 366/ 99/ 233/ 61/ 240/ 110/ 379/
3 6 15 10 16 2 13 3 5 4 9 16 10 8 11 5 3 13 2 16 10 15 6 3 5 11
8 10 16 9 4 5 13 12 1 8 9 7 12 6 11 14 7 2 15 1 14 4 6 12 7 9 8 1
12 13 4 14 1 15 2 7 14 11 2 7 14 11 4 14 1 15 8 1 12 13 6 12 7 9 15
1 14 4 11 14 7 2 9 7 12 6 13 12 1 8 16 9 4 5 5 11 8 10 10 15 6 3 3
13 2 16 10 8 11 5 5 4 9 16 16 2 13 3 3 6 15 10
381/ 37/ 264/ 104/ 372/ 97/ 257/ 28/
16 9 4 5 2 13 3 16 11 14 7 2 5 10 8 11 16 3 13 2 5 4 9 16 11 8
10 5 2 7 14 11 3 6 15 10 7 12 6 9 8 1 12 13 4 15 1 14 9 6 12 7 10
15 6 3 14 1 15 4 13 12 1 8 13 12 1 8 14 1 15 4 10 15 6 3 9 6 12 7 4
15 1 14 8 1 12 13 7 12 6 9 3 6 15 10 2 7 14 11 11 8 10 5 5 4 9 16
16 3 13 2 5 10 8 11 11 14 7 2 2 13 3 16 16 9 4 5
30/ 300/ 169/ 251/ 43/ 262/ 176/ 309/
2 7 14 11 13 3 16 2 8 1 12 13 11 5 10 8 2 16 3 13 11 14 7 2 8 10
5 11 13 12 1 8 16 9 4 5 12 6 9 7 10 15 6 3 14 4 15 1 7 9 6 12 5 4 9
16 1 15 4 14 3 6 15 10 3 6 15 10 1 15 4 14 5 4 9 16 7 9 6 12 14 4
15 1 10 15 6 3 12 6 9 7 16 9 4 5 13 12 1 8 8 10 5 11 11 14 7 2 2 16
3 13 11 5 10 8 8 1 12 13 13 3 16 2 2 7 14 11
192/ 224/ 351/ 15/ 185/ 10/ 346/ 217/
8 13 12 1 10 5 11 8 15 6 3 10 1 14 4 15 8 11 5 10 1 12 13 8 15 4
14 1 10 3 6 15 11 2 7 14 3 16 2 13 4 9 16 5 12 7 9 6 13 2 16 3 14 7
2 11 6 9 7 12 5 16 9 4 5 16 9 4 6 9 7 12 14 7 2 11 13 2 16 3 12 7 9
6 4 9 16 5 3 16 2 13 11 2 7 14 10 3 6 15 15 4 14 1 1 12 13 8 8 11 5
10 1 14 4 15 15 6 3 10 10 5 11 8 8 13 12 1
219/ 113/ 82/ 340/ 230/ 349/ 87/ 120/
10 3 6 15 5 11 8 10 4 9 16 5 15 1 14 4 10 8 11 5 15 6 3 10 4 14
1 15 5 16 9 4 8 13 12 1 16 2 13 3 14 7 2 11 6 12 7 9 3 13 2 16 1 12
13 8 9 7 12 6 11 2 7 14 11 2 7 14 9 7 12 6 1 12 13 8 3 13 2 16 6 12
7 9 14 7 2 11 16 2 13 3 8 13 12 1 5 16 9 4 4 14 1 15 15 6 3 10 10 8
11 5 15 1 14 4 4 9 16 5 5 11 8 10 10 3 6 15
118/ 254/ 325/ 93/ 107/ 84/ 316/ 243/
5 16 9 4 11 8 10 5 14 7 2 11 4 15 1 14 5 10 8 11 4 9 16 5 14 1
15 4 11 2 7 14 10 3 6 15 2 13 3 16 1 12 13 8 9 6 12 7 16 3 13 2 15
6 3 10 7 12 6 9 8 13 12 1 8 13 12 1 7 12 6 9 15 6 3 10 16 3 13 2 9
6 12 7 1 12 13 8 2 13 3 16 10 3 6 15 11 2 7 14 14 1 15 4 4 9 16 5 5
10 8 11 4 15 1 14 14 7 2 11 11 8 10 5 5 16 9 4
241/ 179/ 12/ 322/ 248/ 327/ 21/ 190/
11 2 7 14 8 10 5 11 1 12 13 8 14 4 15 1 11 5 10 8 14 7 2 11 1 15
4 14 8 13 12 1 5 16 9 4 13 3 16 2 15 6 3 10 7 9 6 12 2 16 3 13 4 9
16 5 12 6 9 7 10 3 6 15 10 3 6 15 12 6 9 7 4 9 16 5 2 16 3 13 7 9 6
12 15 6 3 10 13 3 16 2 5 16 9 4 8 13 12 1 1 15 4 14 14 7 2 11 11 5
10 8 14 4 15 1 1 12 13 8 8 10 5 11 11 2 7 14
-
24
* Monitor List of Correspondence between Old and New * ??: 0 1:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 25: 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 73: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
97: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 121: 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 145: 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 169: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 193: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 217:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 241: 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 265: 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 289: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 313: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 337: 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 361: 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[OK!]
The last beautiful result of Monitor list tells us of our
success in reconstructing all 384 primitive solutions from the
Fundamental Three. We could make no other type of solutions, nor
lose any one we expected at all. The Fundamental Three prove to be
really the representative solutions of all. We know what the
solution counts in three levels really mean: 3x16x8=48x8=384. You
may be surprised at the similarity between a self-complementary
type and a pan-diagonal magic one. Yes. There are so many similar
properties between them. #8. Type Conversion between S-C and P-D I
must mention now about the possibility of type conversion here. You
can indeed transform one type into another by some simple line
exchanges. Take your careful look at the next figures. [Figure 5:
How to Transform One into the Other Type] (Step 1) (Step 2) GOAL!
Self-complementary (iii-iv)Exchange (III-IV)Exchange .-----------.
.-----------. .-----------. I |n1|n2|n3|n4| I |n1|n2|n4|n3| I
|n1|n2|n4|n3| |--+--+--+--| |--+--+--+--| |--+--+--+--| II
|n5|n6|n7|n8| II |n5|n6|n8|n7| II |n5|n6|n8|n7| |--+--+--+--|
|--+--+--+--| |--+--+--+--| III |n8|n7|n6|n5| III |n8|n7|n5|n6| IV
|n4|n3|n1|n2| |--+--+--+--| |--+--+--+--| |--+--+--+--| IV
|n4|n3|n2|n1| IV |n4|n3|n1|n2| III |n8|n7|n5|n6| '-----------'
'-----------' '-----------' i ii iii iv i ii iv iii i ii iv iii
(Step 1) (Step 2) GOAL! Pan-diagonal (iii-iv)Exchange
(III-IV)Exchange .-----------. .-----------. .-----------. I
|n1|n2|n3|n4| I |n1|n2|n4|n3| I |n1|n2|n4|n3| |--+--+--+--|
|--+--+--+--| |--+--+--+--| II |n5|n6|n7|n8| II |n5|n6|n8|n7| II
|n5|n6|n8|n7| |--+--+--+--| |--+--+--+--| |--+--+--+--| III
|n3|n4|n1|n2| III |n3|n4|n2|n1| IV |n7|n8|n6|n5| |--+--+--+--|
|--+--+--+--| |--+--+--+--| IV |n7|n8|n5|n6| IV |n7|n8|n6|n5| III
|n3|n4|n2|n1| '-----------' '-----------' '-----------' i ii iii iv
i ii iv iii i ii iv iii
-
25
The step-figures above show how to transform a
self-complementary type into a pan-magic one. Exchange the 3rd4th
rows and columns at the same time. Watch the result and find all
complementary pairs are located only on pan-diagonals. The lower
step-figures show how to transform a pan-diagonal magic type back
into a self-complementary one. Exchange the 3rd4th rows and columns
at the same time. Watch the result and find all complementary pairs
are located symmetrically.
Though each result is not completely the same with the other
start form, it is still a meaningful transformation. The solution
counts of those two types are also the same. We may well accept the
one-to-one correspondence should exist between them. *** Comparison
between 'Self-Complementary' and 'Pan-diagonal' Types: *** ** List
of the Fundamental Three Solutions of Magic Squares of Order 4 **
S/1/P S/2/P S/3/P .--.--.--.--. .--.--.--.--. .--.--.--.--.
.--.--.--.--. .--.--.--.--. .--.--.--.--. | 1| 8|15|10| | 1|
8|10|15| | 1| 8|14|11| | 1| 8|11|14| | 1| 8|12|13| | 1| 8|13|12|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |12|13| 6| 3| |12|13| 3| 6| |12|13| 7|
2| |12|13| 2| 7| |14|11| 7| 2| |14|11| 2| 7| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |14|11| 4| 5| | 7| 2|16| 9| |15|10| 4| 5| | 6| 3|16|
9| |15|10| 6| 3| | 4| 5|16| 9| |--+--+--+--| |--+--+--+--|
|--+--+--+--| |--+--+--+--| |--+--+--+--| |--+--+--+--| | 7| 2|
9|16| |14|11| 5| 4| | 6| 3| 9|16| |15|10| 5| 4| | 4| 5| 9|16|
|15|10| 3| 6| '--'--'--'--' '--'--'--'--' '--'--'--'--'
'--'--'--'--' '--'--'--'--' '--'--'--'--'
[Count = S/3:3/P] OK! #9. Is One-to-one Correspondence broken
between the Standard Solution Sets? But, I noticed the one-to-one
correspondence is broken between the two sets of 48 Standard
Solutions afterward. Look carefully at the next list shown below.
** Transformation of MS44: Between 'Self-Complementary' and
'Complete' ** ** Comparison between /Self-Complementary and
/Complete Types ** S/ 1/C S/ 2/C S/ 3/C 1 8 15 10 1 8 10 15 1 8 14
11 1 8 11 14 1 8 15 10 1 8 10 15 12 13 6 3 12 13 3 6 12 13 7 2 12
13 2 7 14 11 4 5 14 11 5 4 14 11 4 5 7 2 16 9 15 10 4 5 6 3 16 9 12
13 6 3 7 2 16 9 7 2 9 16 14 11 5 4 6 3 9 16 15 10 5 4 7 2 9 16 12
13 3 6
S/ 4/C S/ 5/C S/ 6/C 1 8 12 13 1 8 13 12 1 8 14 11 1 8 11 14 1 8
12 13 1 8 13 12 14 11 7 2 14 11 2 7 15 10 4 5 15 10 5 4 15 10 6 3
15 10 3 6 15 10 6 3 4 5 16 9 12 13 7 2 6 3 16 9 14 11 7 2 4 5 16 9
4 5 9 16 15 10 3 6 6 3 9 16 12 13 2 7 4 5 9 16 14 11 2 7
S/ 7/C S/ 8/C S/ 9/C 1 12 15 6 1 12 6 15 1 12 8 13 1 12 13 8 1
12 14 7 1 12 7 14 14 7 4 9 14 7 9 4 14 7 11 2 14 7 2 11 15 6 4 9 15
6 9 4 8 13 10 3 11 2 16 5 15 6 10 3 4 9 16 5 8 13 11 2 10 3 16 5 11
2 5 16 8 13 3 10 4 9 5 16 15 6 3 10 10 3 5 16 8 13 2 11
S/10/C S/11/C S/12/C 1 12 8 13 1 12 13 8 1 14 12 7 1 14 7 12 1
14 8 11 1 14 11 8 15 6 10 3 15 6 3 10 15 4 6 9 15 4 9 6 15 4 10 5
15 4 5 10 14 7 11 2 4 9 16 5 8 11 13 2 10 5 16 3 12 7 13 2 6 9 16 3
4 9 5 16 14 7 2 11 10 5 3 16 8 11 2 13 6 9 3 16 12 7 2 13
S/13/C S/14/C S/15/C 2 7 16 9 2 7 9 16 2 7 13 12 2 7 12 13 2 7
16 9 2 7 9 16 11 14 5 4 11 14 4 5 11 14 8 1 11 14 1 8 13 12 3 6 13
12 6 3 13 12 3 6 8 1 15 10 16 9 3 6 5 4 15 10 11 14 5 4 8 1 15 10 8
1 10 15 13 12 6 3 5 4 10 15 16 9 6 3 8 1 10 15 11 14 4 5
-
26
S/16/C S/17/C S/18/C
2 7 11 14 2 7 14 11 2 7 13 12 2 7 12 13 2 7 11 14 2 7 14 11 13
12 8 1 13 12 1 8 16 9 3 6 16 9 6 3 16 9 5 4 16 9 4 5 16 9 5 4 3 6
15 10 11 14 8 1 5 4 15 10 13 12 8 1 3 6 15 10 3 6 10 15 16 9 4 5 5
4 10 15 11 14 1 8 3 6 10 15 13 12 1 8
S/19/C S/20/C S/21/C 2 11 16 5 2 11 5 16 2 11 7 14 2 11 14 7 2
11 13 8 2 11 8 13 13 8 3 10 13 8 10 3 13 8 12 1 13 8 1 12 16 5 3 10
16 5 10 3 7 14 9 4 12 1 15 6 16 5 9 4 3 10 15 6 7 14 12 1 9 4 15 6
12 1 6 15 7 14 4 9 3 10 6 15 16 5 4 9 9 4 6 15 7 14 1 12
S/22/C S/23/C S/24/C
2 11 7 14 2 11 14 7 2 13 11 8 2 13 8 11 2 13 7 12 2 13 12 7 16 5
9 4 16 5 4 9 16 3 5 10 16 3 10 5 16 3 9 6 16 3 6 9 13 8 12 1 3 10
15 6 7 12 14 1 9 6 15 4 11 8 14 1 5 10 15 4 3 10 6 15 13 8 1 12 9 6
4 15 7 12 1 14 5 10 4 15 11 8 1 14
25/S 0/T 25/C 26/S 0/T 26/C 3 6 16 9 3 6 9 16 3 6 9 16 3 6 13 12
3 6 12 13 3 6 15 10 10 15 5 4 10 15 4 5 13 12 7 2 10 15 8 1 10 15 1
8 13 12 1 8 13 12 2 7 8 1 14 11 8 1 14 11 16 9 2 7 5 4 14 11 2 7 14
11 8 1 11 14 13 12 7 2 10 15 4 5 5 4 11 14 16 9 7 2 16 9 4 5
27/S 25/T 27/C 28/S 27/T 28/C 3 6 16 9 3 6 9 16 3 6 12 13 3 6 13
12 3 6 12 13 3 6 15 10 13 12 2 7 13 12 7 2 16 9 7 2 16 9 2 7 16 9 7
2 16 9 4 5 10 15 5 4 8 1 14 11 5 4 14 11 10 15 8 1 5 4 14 11 2 7 14
11 8 1 11 14 10 15 4 5 10 15 1 8 5 4 11 14 10 15 1 8 13 12 1 8
29/S 29/T 29/C 30/S 31/T 30/C
3 10 16 5 3 10 5 16 3 10 5 16 3 10 13 8 3 10 8 13 3 10 15 6 13 8
2 11 13 8 11 2 13 8 11 2 16 5 2 11 16 5 11 2 13 8 1 12 6 15 9 4 12
1 14 7 12 1 14 7 6 15 12 1 9 4 14 7 2 11 14 7 12 1 7 14 6 15 4 9 6
15 4 9 9 4 7 14 6 15 1 12 16 5 4 9
31/S 33/T 31/C 32/S 34/T 32/C 3 13 10 8 3 13 8 10 3 10 8 13 3 13
6 12 3 13 12 6 3 10 15 6 16 2 5 11 16 2 11 5 16 5 11 2 16 2 9 7 16
2 7 9 16 5 4 9 6 12 15 1 9 7 14 4 9 4 14 7 10 8 15 1 5 11 14 4 2 11
14 7 9 7 4 14 6 12 1 15 6 15 1 12 5 11 4 14 10 8 1 15 13 8 1 12
33/S 0/T 33/C 34/S 0/T 34/C
4 5 15 10 4 5 10 15 3 13 8 10 4 5 14 11 4 5 11 14 3 13 12 6 9 16
6 3 9 16 3 6 16 2 11 5 9 16 7 2 9 16 2 7 16 2 7 9 14 11 1 8 7 2 13
12 9 7 14 4 15 10 1 8 6 3 13 12 5 11 14 4 7 2 12 13 14 11 8 1 6 12
1 15 6 3 12 13 15 10 8 1 10 8 1 15
35/S 35/T 35/C 36/S 37/T 36/C 4 5 15 10 4 5 10 15 4 5 10 15 4 5
14 11 4 5 11 14 4 5 16 9 14 11 1 8 14 11 8 1 14 11 8 1 15 10 1 8 15
10 8 1 14 11 2 7 9 16 6 3 7 2 13 12 7 2 13 12 9 16 7 2 6 3 13 12 1
8 13 12 7 2 12 13 9 16 3 6 9 16 3 6 6 3 12 13 9 16 2 7 15 10 3
6
37/S 39/T 37/C 38/S 41/T 38/C
4 9 15 6 4 9 6 15 4 5 11 14 4 9 14 7 4 9 7 14 4 5 16 9 14 7 1 12
14 7 12 1 15 10 8 1 15 6 1 12 15 6 12 1 15 10 3 6 5 16 10 3 11 2 13
8 6 3 13 12 5 16 11 2 10 3 13 8 1 8 13 12 11 2 8 13 5 16 3 10 9 16
2 7 10 3 8 13 5 16 2 11 14 11 2 7
39/S 43/T 39/C 40/S 44/T 40/C 4 14 9 7 4 14 7 9 4 9 6 15 4 14 5
11 4 14 11 5 4 9 16 5 15 1 6 12 15 1 12 6 14 7 12 1 15 1 10 8 15 1
8 10 14 7 2 11 5 11 16 2 10 8 13 3 11 2 13 8 9 7 16 2 6 12 13 3 1
12 13 8 10 8 3 13 5 11 2 16 5 16 3 10 6 12 3 13 9 7 2 16 15 6 3
10
-
27
41/S 0/T 41/C 42/S 0/T 42/C
5 4 16 9 5 4 9 16 4 9 7 14 5 4 16 9 5 4 9 16 4 9 16 5 10 15 3 6
10 15 6 3 15 6 12 1 11 14 2 7 11 14 7 2 15 6 3 10 11 14 2 7 8 1 12
13 10 3 13 8 10 15 3 6 8 1 12 13 1 12 13 8 8 1 13 12 11 14 7 2 5 16
2 11 8 1 13 12 10 15 6 3 14 7 2 11
43/S 0/T 43/C 44/S 0/T 44/C 5 10 11 8 5 10 8 11 4 14 7 9 5 11 10
8 5 11 8 10 4 14 11 5 16 3 2 13 16 3 13 2 15 1 12 6 16 2 3 13 16 2
13 3 15 1 8 10 4 15 14 1 9 6 12 7 10 8 13 3 4 14 15 1 9 7 12 6 6 12
13 3 9 6 7 12 4 15 1 14 5 11 2 16 9 7 6 12 4 14 1 15 9 7 2 16
45/S 0/T 45/C 46/S 0/T 46/C
6 3 15 10 6 3 10 15 5 4 14 11 6 3 15 10 6 3 10 15 5 4 15 10 9 16
4 5 9 16 5 4 16 9 7 2 12 13 1 8 12 13 8 1 16 9 6 3 12 13 1 8 7 2 11
14 3 6 12 13 9 16 4 5 7 2 11 14 2 7 12 13 7 2 14 11 12 13 8 1 10 15
1 8 7 2 14 11 9 16 5 4 11 14 1 8
47/S 0/T 47/C 48/S 0/T 48/C 6 9 12 7 6 9 7 12 6 3 13 12 6 12 9 7
6 12 7 9 6 3 16 9 15 4 1 14 15 4 14 1 15 10 8 1 15 1 4 14 15 1 14 4
15 10 5 4 3 16 13 2 10 5 11 8 4 5 11 14 3 13 16 2 10 8 11 5 1 8 11
14 10 5 8 11 3 16 2 13 9 16 2 7 10 8 5 11 3 13 2 16 12 13 2 7 *
Monitor List of Solution Correspondence between /T and /C * ??: 12
1: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 25: 1 0 1 0 1 0
1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0
[Count = S/48:48/C]